Fractal Analysis for Natural Hazards - Special Publication No 261 (Geological Society Special Publication)

Fractal Analysis for Natural Hazards The Geological Society of London Books Editorial Committee B. PANKHURST (UK) (CH...

Author: G. Cello; B. D. Malamud; Editors

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Fractal Analysis for Natural Hazards

The Geological Society of London Books Editorial Committee B. PANKHURST (UK) (CHIEF EDITOR)

Society Books Editors J. GREGORY (UK) J. GRIFFITHS (UK) J. HOWE (UK) P. LEAT (UK) N. ROBINS (UK) J. TURNER (UK)

Society Books Advisors M. BROWN (USA) R. GIERI~ ( G e r m a n y ) J. GLUYAS (UK) D. STEAD (Canada) R. STEPHENSON (Netherlands) S. TURNER (Australia)

Geological Society books refereeing procedures The Society makes every effort to ensure that the scientific and production quality of its books matches that of its journals. Since 1997, all book proposals have been refereed by specialist reviewers as well as by the Society's Books Editorial Committee. If the referees identify weaknesses in the proposal, these must be addressed before the proposal is accepted. Once the book is accepted, the Society Book Editors ensure that the volume editors follow strict guidelines on refereeing and quality control. We insist that individual papers can only be accepted after satisfactory review by two independent referees. The questions on the review forms are similar to those for Journal of the Geological Society. The referees' forms and comments must be available to the Society's Book Editors on request. Although many of the books result from meetings, the editors are expected to commission papers that were not presented at the meeting to ensure that the book provides a balanced coverage of the subject. Being accepted for presentation at the meeting does not guarantee inclusion in the book. More information about submitting a proposal and producing a book for the Society can be found on its web site: www.geolsoc.org.uk.

It is recommended that reference to all or part of this book should be made in one of the following ways: CELLO, G. & MALAMUD,B. D. (eds) 2006. Fractal Analysis for Natural Hazards. Geological Society, London, Special Publications, 261. KIDSON, R., RICHARDS, K. S. & CARLING, P. A. 2006. Power-law extreme flood frequency. In: CELLO, G. & MALAMUD, B. D. (eds) Fractal Analysis for Natural Hazards. Geological Society, London, Special Publications, 261, 141-153.

GEOLOGICAL SOCIETY SPECIAL PUBLICATION NO. 261

Fractal Analysis for Natural Hazards EDITED BY

G. CELLO University of Camerino, Italy and

B. D. MALAMUD King's College London, UK

2006 Published by The Geological Society London

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Contents

Preface

MALAMUD,B. D. • TURCOTTE,D. L. An inverse cascade explanation for the power-law

vii 1

frequency-area statistics of earthquakes, landslides and wildfires LEI, X. Typical phases of pre-failure damage in granitic rocks under differential compres slon

11

DAVY, P., BOUR, O., DE DREUZY, J.-R. & DARCEL, C. Flow in multiscale fractal fracture networks

31

PAPARO, G., GREGORI,G. P., POSCOLIERI, M., MARSON, I., ANGELUCCI,F. &

47

GLORIOSO, G. Crustal stress crises and seismic activity in the Italian peninsula investigated by fractal analysis of acoustic emission, soil exhalation and seismic data POSCOLIERI, M., LAGIOS, E., GREGORI, G. P., PAPARO, G., SAKKAS, V.A., PARCHARIDIS, I., MARSON, I., SOUKIS, K., VASSILAKIS,E., ANGELUCCI, F. & VASSmOPOULOtJ, S. Crustal stress and seismic activity in the Ionian archipelago as inferred by satellite- and ground-based observations, Kefallin]a, Greece

63

ZVELEBIL, J., PALU~, M. & NOVOTNA, D. Nonlinear Science issues in the dynamics of unstable rock slopes: new tools for rock fall risk assessment and early warnings

79

TELESCA, L., LAPENNA, V. & MACCHIATO,M. Multifractal variability in self-potential signals measured in seismic areas

95

TURCOTTE, D. L., MALAMUD, B. D., GUZZETTI, F. 8~ REICHENBACH,P. A general landslide distribution applied to a small inventory in Todi, Italy

105

MARCHEGIANI, L., VAN DIJK, J. P., GILLESPIE, P. A., TONDI, E. & CELLO, G. Scaling properties of the dimensional and spatial characteristics of fault and fracture systems in the Majella Mountain, central Italy

113

CELLO, G., MARCHEGIAM,L. & TONDI, E. Evidence for the existence of a simple relation between earthquake magnitude and the fractal dimension of seismogenic faults: a case study from central Italy

133

KIDSON, R., RICHARDS, K. S. & CARLING, P. A. Power-law extreme flood frequency

141

MmL~Nra'ON, J. D. A., PERRY, G. L. W. & MALAMUD, B. D. Models, data and mechanisms: quantifying wildfire regimes

155

Index

169

Preface

Self-similarity and fractals, the idea that an object's pattern will approximately repeat itself at multiple scales, was conceptualized and formalized by the Polish-born French scientist Benoit Mandelbrot, in his pioneering 1967 Science paper 'How long is the coast of England: statistical self-similarity and fractional dimension'. Since this publication, over 22,000 peer-review papers have been published in the broad social and physical sciences that use the ideas of fractals, along with the publication of many hundreds of books on the subject. In the Earth Sciences, the concept of self-similar scaling (scale invariance) and fractal geometry over a given range of scales is well recognized in many natural objects, for example sand dunes, rock fractures and folds, and drainage networks. However, the use of fractals for spatial and temporal analyses has been less used (and accepted) in the Earth Sciences, particularly for the study of natural hazards. This book brings together twelve contributions that emphasize the role of fractal analyses in natural hazard research, with the

papers based on a scientific session at the 32nd International Geological Congress held in Firenze, Italy, August 2004. The main natural hazards discussed in these papers include landslides and rock falls, wildfires, floods, catastrophic rock fractures and earthquakes. A wide variety of spatial and temporal fractal-related approaches and techniques are applied to 'natural' data, experimental data, and computer simulations. These approaches and techniques include probabilistic hazard analysis, cellular-automata models, spatial analyses, temporal variability, prediction, and concepts such as self-organizing behaviour. The main aims of this volume of papers are (a) to present current research on fractal analyses as applied to natural hazards, and (b) to stimulate the curiosity of advanced Earth Science students and researchers in the use of fractal analyses for the better understanding of natural hazards.

GIUSEPPE CELLO and BRUCE D. MALAMUD

Some words from the colleagues and friends of Giuseppe Cello: On the 5th of July 2006, during the editing process of this book, Giuseppe Cello passed away due to heart failure. Of the scientist, we will always remember many useful discussions, constructive suggestions, and his innovative approach to the broad areas of Structural Geology. Of the man, we will never forget his genuine friendship, his sincerity and his love for every little thing in life. Peppe's memory will remain indelible in all of us.

Contents

Preface

MALAMUD,B. D. • TURCOTTE,D. L. An inverse cascade explanation for the power-law

vii 1

frequency-area statistics of earthquakes, landslides and wildfires LEI, X. Typical phases of pre-failure damage in granitic rocks under differential compres slon

11

DAVY, P., BOUR, O., DE DREUZY, J.-R. & DARCEL, C. Flow in multiscale fractal fracture networks

31

PAPARO, G., GREGORI,G. P., POSCOLIERI, M., MARSON, I., ANGELUCCI,F. &

47

GLORIOSO, G. Crustal stress crises and seismic activity in the Italian peninsula investigated by fractal analysis of acoustic emission, soil exhalation and seismic data POSCOLIERI, M., LAGIOS, E., GREGORI, G. P., PAPARO, G., SAKKAS, V.A., PARCHARIDIS, I., MARSON, I., SOUKIS, K., VASSILAKIS,E., ANGELUCCI, F. & VASSmOPOULOtJ, S. Crustal stress and seismic activity in the Ionian archipelago as inferred by satellite- and ground-based observations, Kefallin]a, Greece

63

ZVELEBIL, J., PALU~, M. & NOVOTNA, D. Nonlinear Science issues in the dynamics of unstable rock slopes: new tools for rock fall risk assessment and early warnings

79

TELESCA, L., LAPENNA, V. & MACCHIATO,M. Multifractal variability in self-potential signals measured in seismic areas

95

TURCOTTE, D. L., MALAMUD, B. D., GUZZETTI, F. 8~ REICHENBACH,P. A general landslide distribution applied to a small inventory in Todi, Italy

105

MARCHEGIANI, L., VAN DIJK, J. P., GILLESPIE, P. A., TONDI, E. & CELLO, G. Scaling properties of the dimensional and spatial characteristics of fault and fracture systems in the Majella Mountain, central Italy

113

CELLO, G., MARCHEGIAM,L. & TONDI, E. Evidence for the existence of a simple relation between earthquake magnitude and the fractal dimension of seismogenic faults: a case study from central Italy

133

KIDSON, R., RICHARDS, K. S. & CARLING, P. A. Power-law extreme flood frequency

141

MmL~Nra'ON, J. D. A., PERRY, G. L. W. & MALAMUD, B. D. Models, data and mechanisms: quantifying wildfire regimes

155

Index

169

An inverse cascade explanation for the power-law frequency-area statistics of earthquakes, landslides and wildfires B R U C E D. M A L A M U D t & D O N A L D L. T U R C O T T E 2

1Environmental Monitoring and Modelling Research Group, Department of Geography, King's College London, Strand, London WC2R 2LS, UK (e-mail: [email protected]) 2Department of Geology, University of California, Davis, CA, 95616, USA (e-mail: turcotte@ geology.ucdavis.edu) Abstract: Frequency-magnitude statistics for natural hazards can greatly help in probabilistic hazard assessments. An example is the case of earthquakes, where the generality of a power-law (fractal) frequency-rupture area correlation is a major feature in seismic risk mapping. Other examples of this power-law frequency-size behaviour are landslides and wildfires. In previous studies, authors have made the potential association of the hazard statistics with a simple cellular-automata model that also has robust power-law statistics: earthquakes with slider-block models, landslides with sandpile models, and wildfires with forest-fire models. A potential explanation for the robust power-law behaviour of both the models and natural hazards can be made in terms of an inverse-cascade of metastable regions. A metastable region is the region over which an 'avalanche' spreads once triggered. Clusters grow primarily by coalescence. Growth dominates over losses except for the very largest clusters. The cascade of cluster growth is self-similar and the frequency of cluster areas exhibits power-law scaling. We show how the power-law exponent of the frequency-area distribution of clusters is related to the fractal dimension of cluster shapes.

The f r e q u e n c y - size statistics of a number of natural hazards appear to satisfy power-law (fractal) scaling to a good approximation under a wide variety of conditions (for a review, see Malamud 2004). These include earthquakes (Aki 1981; Turcotte & Malamud 2002), volcanic eruptions (Pyle 2000), wildfires (Malamud et al. 1998, 2005; Ricotta et al. 1999), landslides (Guzzetti et al. 2002; Malamud et al. 2004), asteroid impacts (Chapman & Morrison 1994; Chapman 2004) and potentially floods (Turcotte & Greene 1993; Turcotte 1994; Malamud et al. 1996; Malamud & Turcotte 2006). In this paper, we will consider the frequency-area statistics of three of these natural hazards: earthquakes, landslides, and wildfires. In each case the 'noncumulative' number of events N with area A satisfies the power-law relation N ~ A -/3

(1)

to a good approximation and over many orders of magnitude, where/3 is a constant. A number of simple cellular-automata models have also been shown to exhibit robust power-law behaviour (e.g. see Malamud & Turcotte 2000), including the slider-block, sandpile, and forest-fire models. In this paper we will first discuss the f r e q u e n c y area statistics of earthquakes, landslides, and

wildfires. We will then discuss the f r e q u e n c y area statistics of the slider-block, sandpile, and forest-fire cellular-automata models. Finally, in the context of real wildfires and the forest-fire model, we will advance a potential explanation for the robust power-law behaviour of both in terms of an inverse-cascade of metastable regions.

Earthquakes The first natural hazard that was recognized to exhibit power-law frequency-area statistics was earthquakes. For more than fifty years it has been accepted that the rate at which earthquakes occur in a region generally satisfies the G u t e n b e r g Richter (1954) frequency-magnitude relation log ]//CE = --bM + log h

(2)

where/r is the cumulative number of earthquakes with a magnitude greater than or equal to M in a specified area and time, and b and h are constants. Aki (1981) showed that Eq. (2) is equivalent to the power-law relation: NcE ~ AE-(t~-l)

(3)

where AE is the earthquake rupture area and (/3 - 1) = b in Eq. (2). The equivalent frequency-

From: CELLO, G. & MALAMUD,B. D. (eds) 2006. FractalAnalysisfor Natural Hazards. Geological Society, London, Special Publications, 261, 1-9. 0305-8719/06/$15.00 9 The Geological Society of London 2006.

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Fig. 1. Cumulative frequency statistics of earthquakes in southern California (figure after Rundle et al. 2003). Shown are the cumulative number of earthquakes, NCE, occurring in southern California, with magnitudes greater than or equal to M as a function of M. Also shown is the equivalent square root of the rupture area, AlE/2. Twenty individual years are considered, with data from the Southern California Seismic Network: (a) 1983-1987, (b) 1988-1992, (c) 1993-1997, (d) 1998-2002. The solid straight lines in (a) to (d) are the Gutenberg-Richter powerlaw relation (Eq. 2) with b = 1.0 and ti = 2.5 x 105 yr -1, fit to all the data from 1983-2002. The larger number of earthquakes in 1987, 1992, 1994, and 1999 can be attributed to the aftershocks of the Whittier-Narrows, Landers, Northridge, and Hector Mine earthquakes, respectively. During the other sixteen years, when there are not a large number of aftershocks, the data closely represent the background seismicity in southern California, and are nearly uniform from year to year, with very similar power-law exponents.

area distribution for Eq. (3) that is noncumulative has a power-law exponent of /3, the same as in Eq. (1). As just one example of Gutenberg-Richter scaling, in Figure 1 we consider regional seismicity in southern California on a yearly basis, as given by Rundle et al. (2003). Plotted for each individual year in the period 1983-2002 are the cumulative numbers of earthquakes ]VcE with magnitudes greater than or equal to M as a function of M. Also include is the Gutenberg-Richter power-law relation (Eq. 2) (fit to all data, 1983-2002) with b=l.0andd= 2 . 5 x 105 yr -], shown as solid straight lines in Figures l a - d . There is generally good agreement between each individual year's data and the Gutenberg-Richter relation (solid

line) for the whole period of record. The exceptions can be attributed to the aftershock sequences of the 1987 Whittier-Narrows, 1992 Landers, 1994 Northridge, and 1999 Hector Mine earthquakes. During the sixteen other years when there are not a large number of aftershocks, the data closely represent the background seismicity in southern California, and are nearly uniform from year to year, with very similar power-law exponents. Earthquakes have been shown to satisfy powerlaw frequency-size statistics over many regions around the world and over many orders of earthquake magnitude (Frohlich & Davis 1993; Kossobokov et al. 2000), although some authors question the validity for the very largest earthquakes (Scholz 1997).

INVERSE CASCADE AND NATURAL HAZARDS

Landslides The second natural hazard we consider is the landslide, a complex natural phenomenon that constitute a serious hazard in many countries (see also Turcotte et al. 2006 in this volume, which discusses landslide statistics in much more detail). Landslides are generally the result of triggers such as intense rainfall or earthquakes, with the trigger resulting in a landslide event that may include anything from just one single landslide up to many tens of thousands. A triggered landslide event can be quantified using landslide inventories, which normally includes a tabulation of landslide areas, spatial position, and landslide type. Malamud et al. (2004) considered the frequencyarea statistics of three 'fresh' triggered landslide events. The inventories for each landslide event were substantially complete, consisting of (1) more than 11,000 landslides triggered by the 17 January 1994 Northridge (California) earthquake; (2) more than 4000 landslides triggered by a rapid snowmelt event in the Umbria region of Italy in January 1997; and (3) more than 9000 landslides triggered by heavy rainfall in Guatemala during late October and early November 1998. They considered the probability densities p(AL) of landslide areas AL, defined as p(AL) --

1

6NL

NLT 8AL

(4)

where ~NL is the number of landslides with areas between AL and AL q-~AL, and NLT is the total number of landslides in the inventory. Malamud et al. (2004) showed that the three sets of probability densities were in excellent agreement with each other, and correlated well with a threeparameter inverse gamma probability distribution function (pdf), which for medium and large landslide areas can be approximated by

p(AL) "-~ aF(/3 - 1)

(5)

3

considered by Malamud et al. (2004) contained 4000-11,000 landslides. To examine further the 'general' landslide distribution given by Eq. (5) Turcotte et al. (2006) (in this volume) examined a much smaller, but still substantially complete, inventory of 165 landslides triggered by rainfall in the region of Todi, Central Italy, and found a very similar power-law exponent/3. There is accumulating evidence (for a review, see Guzzetti et al. 2002) that the frequency-area distribution of medium and large landslides decays as an inverse power of the landslide area. This behaviour is observed despite large differences in landslide types, sizes, distributions, patterns, and triggering mechanisms.

Wildfires Our final example of a natural hazard that follows power-law frequency-area statistics to a good approximation is the wildfire. Malamud et al. (1998) considered four wildfire data sets from the USA and Australia. The four data sets come from a wide variety of geographic regions with different vegetation types and climates. In each case, the noncumulative number of fires per year plotted as a function of burned fire area AF correlated well with the power-law relationship (Eq. 1), with/3 = 1.3-1.5. Malamud et al. (2005) carried out a comprehensive study of the frequency-area statistics of 88,916 wildfires on United States Forest Service lands in the conterminous USA during the period 1970-2000, examining the statistics both spatially, as a function of ecoregion, and temporally. Ecoregions are land units classified by climate, vegetation, and topography. As the wildfire inventories used were not 'complete' (there are many more 'smaller' wildfires than measured), probability densities as defined in Eq. (4) were not appropriate for the analyses and they used frequency densities f(AF) defined as

6NF f(AF) -- 6AF

where AL is the area of individual landslides, a and 13 are constants, and F ( / 3 - 1) is the gamma function of/3 - 1. The tail of the probability distribution for large landslide areas is a power-law with exponent /3, equivalent to Eq. (1). Malamud et al. (2004) used a maximum-likelihood fit of the inverse-gamma distribution to the three sets of landslide probability densities, taking a = 1.28 x 10-3km 2 and /3 = 2.40. Thus the power-law exponent for the medium and large landslides (the 'tail' of the distribution) is /3=2.40. The three inventories

(6)

where AF is the wildfire burned area, and 6NF the number of wildfires in a 'bin' of width 6AF. The frequency densities f(AF) are then the number of wildfires per 'unit' bin. The frequency density f(AF) in Eq. (6) is equal to the probability density p(AF) introduced in Eq. (4) multiplied by the total number of wildfires in the inventory NFT. For each of eighteen different ecoregions examined in the conterminous USA, Malamud et al. (2005) found that the noncumulative number of fires per year plotted as a function of burned fire area AF

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Fig. 2. Normalized frequency-area wildfire statistics for (a) Mediterranean and (b) Subtropical ecoregion divisions, for the period 1970-2000 (figure after Malamud et al. 2005). Shown (circles) are normalized frequency densitiesf(AF) (number of wildfires per 'unit bin' of 1 km 2, normalized by database length in years and USFS area within the ecoregion) plotted as a function of wildfire area AF. Also shown for both ecoregions is a solid line, the best least-squares fit to Eq. (1), with coefficient of determination r 2. Dashed lines represent lower/upper 95% confidence intervals, calculated from the standard error. Horizontal error bars are due to measurement and size binning of individual wildfires. Vertical error bars represent two standard deviations ( ___2 s.d.) of the normalized frequency densitiesj~(AF).

correlated well with the power-law relationship (Eq. 1), with 13 = 1.30-1.81. Two examples are given in Figure 2, showing the two extremes of values obtained by the authors. In Figure 2a are presented the frequency-area statistics for 16,423 wildfires in the Subtropical ecoregion division (within the southeastern part of the USA) and in Figure 2b, 475 wildfires in the Mediterranean ecoregion division (within California, USA). In both cases, excellent correlations are obtained with the power-law relationship (Eq. 1), with /3 = 1.81 _ 0.07 (___2 s.d.) for the Subtropical ecoregion and /3 = 1.30 _ 0.05 ( + 2 s.d.) for the Mediterranean ecoregion. One of the purposes of this paper is to examine this variability of the power-law exponent/3. A number of other authors (e.g. Niklasson & Granstrom 2000; Minnich 2001" Ricotta et al. 2001) have also found good correlations of the frequency-area distributions of wildfires with the power-law relation (1), although others disagree (Cumming 2001; Reed & McKelvey 2002). Millington et al. (2006), another chapter in this volume, give a detailed discussion and review of power-law scaling in wildfire areas. In addition to wildfire areas, some authors (Sole & Manrubia 1995a, b) have found that the frequency-area distributions of clusters of trees in forests also follow power-law statistics. Considering the many complexities of the initiation and propagation of wildfires, it is remarkable that the frequency-area statistics are similar under a wide variety of environments. The

proximity of combustible material varies widely. The behaviour of a particular wildfire depends strongly on meteorological conditions. Fire-fighting efforts extinguish many fires. Despites these complexities, the power-law frequency-area statistics of actual wildfires seems very robust.

Cellular-automata models Cellular-automata (CA) models are lattice-based models that have simple 'rules', but often exhibit complex behaviour. At each time step, a series of nearest-neighbour rules of interaction are applied, and individual cells are updated in the next step. A brief history of cellular automata is given by Sarkar (2000). A number of 'simple' CA models have been shown to exhibit robust power-law statistics (for reviews see Turcotte 1999; Malamud & Turcotte 2000). We will discuss three of these here, the sandpile, forest-fire, and slider-block models. We begin with the sandpile model introduced by B a k et al. (1988) in the context of his discussions on self-organized criticality. In this model there is a square grid of boxes and at each time step a particle is dropped into a randomly selected box. When a box accumulates four particles, they are redistributed to the four adjacent boxes, or in the case of edge boxes, lost from the grid. Redistributions can lead to further instabilities, with 'avalanches' of particles lost from the edges of the grid. Because of this 'avalanche' behaviour, this was called a 'sandpile' model. This is a cellular-automata

INVERSE CASCADE AND NATURAL HAZARDS model because the boxes are the cells and the nearest-neighbour redistribution rules constitute the automata. This CA model, and others like it, generate 'avalanches' with a power-law frequencysize distribution, and contains a steady-state 'input', with the 'output' occurring in the 'avalanches'. The noncumulative frequency-area distribution of model avalanches was found to satisfy the power-law distribution given in Eq. (1) with / 3 = 1.0-1.3 (Bak et al. 1988; Kadanoff et al. 1989; Turcotte 1997). A second CA model with robust power-law statistics is the forest-fire model (Bak et al. 1990; Drossel & Schwabl 1992a, b). In the simplest version of this model, a square grid of sites is considered. At each time step either a tree is planted on a randomly chosen site (if the site is unoccupied) or a spark is dropped on the site. If the spark is dropped on a site with a tree, that tree and all adjacent sites with trees are 'burned' in a model 'forest fire'. The firing frequency f is the inverse number of attempted tree drops on the square grid before a model match is dropped on a randomly chosen site. If f = 1/ 100, there have been 99 attempts to plant trees (some successful, some unsuccessful) before a match is dropped at the 100th time step. Two examples of model fires are given in Figure 3. For a broad range of grid sizes and firing frequencies (discussed in more detail in the next section), the frequency-area distribution of the small and medium model fires again satisfies Eq. (1) with /3 = 1.19 (Grassberger 2002). A third CA model that exhibits robust power-law statistics is the slider-block model (Burridge & Knopoff 1967; Carlson & Langer 1989). In this model, an array of slider blocks are connected to

5

a constant velocity driver plate by puller springs and to each other by connector springs. The blocks exhibit stick-slip behaviour as a result of frictional interactions with the plate across which they are pulled. The area is defined to be the number of blocks that participate in a slip event. The frequency-area distribution of smaller and medium slip events again satisfies Eq. (1), with / 3 = 1.0-1.5 (Carlson & Langer 1989; Huang et al. 1992; Carlson et al. 1994). This model is deterministic, whereas the sandpile and forest-fire models are stochastic. A number of authors have discussed each of these CA models in the context of specific natural hazards: the sandpile model with landslides (e.g. Guzzetti et al. 2002), the forest-fire model with wildfires (Malamud et al. 1998), and the sliderblock model with earthquakes (Burridge & Knopoff 1967; Carlson & Langer 1989). In the next section, we will advance an 'explanation' for both the behaviour of the models and natural hazards in terms of the coalescence of growing metastable regions, by introducing the inversecascade model in the context of the forest-fire model.

Metastable regions and the inverse-cascade model Before introducing the inverse-cascade model, we will discuss the role of metastable regions in CA models. A metastable region is the region over which an 'avalanche' spreads once the region is triggered. The role of metastable regions can be illustrated by the forest-fire model. In this model,

Fig. 3. Two forest-fire model examples using a grid of 128 • 128 cells and a forest-fire run with sparking frequency 1/f = 2000. The black squares constitute the model forest fires. The light grey squares are unburned trees. The white regions are unoccupied grid points (i.e. no trees). The area of the model fire in (a) is AF = 204 trees and in (b) AF = 5237 trees; the latter is seen to span the entire grid.

6

B.D. MALAMUD & D. L. TURCOTTE

a metastable region is a cluster of trees that will burn when any of the trees is ignited by a match. Thus at any one time, all clusters of trees on the grid are metastable in the sense that each one has the potential to be ignited. We will denote as AF the area of a cluster, NF the number of model fires at a given size A F that bum over time, and Ncluste r the number of clusters on the grid (at any one time) with area AF, each of which could potentially burn to give a fire of size A F . Because the probability that a match will strike a cluster is proportional to its a r e a A F , then the probability that a cluster of size AF will burn in a specified time interval, is proportional to the product of the cluster a r e a A F and the number of clusters gcluster, or NF ~AFNclus~r.

(7)

Because the frequency-area distribution satisfies Eq. (1), NF ~ AF-~, it follows from Eq. (7) that Ncluster ~ AF (fl+l).

(8)

A distribution of fire sizes over time is a powerlaw with exponent -/3. This implies a distribution of cluster sizes on the grid at one 'snapshot' in time, that is, a power-law with exponent - (/3 + 1). The inverse-cascade model is a relatively simple model that might provides a potential explanation for the power-law frequency-size distributions found in both the CA models and actual natural hazards. To better understand the inverse-cascade model, we continue our discussion in the context of the forest-fire model. It has been shown (Turcotte et al. 1999; Turcotte 1999; Gabrielov et al. 1999) that tree clusters grow primarily by coalescence. There are three ways that clusters form on the grid: (1)

(2)

(3)

A newly planted tree forms a 'single cluster'; in other words there are no adjacent cells containing trees and a cluster of 1 tree is formed. A tree is planted immediately adjacent to an existing cluster with Ai trees; a new cluster of ( A i § 1) trees is formed. A newly planted tree bridges the gap between two clusters with A i and Aj trees; a new cluster § 1) trees. is formed with (A i §

A detailed study of cluster growth in the forest-fire model by Yakovlev et al. (2005) has shown that the last method, cluster coalescence, dominates the cluster growth process. Trees cascade from smaller to larger clusters until they are lost in the fires that destroy the largest clusters, and the cascade is terminated. This is termed an 'inverse cascade', because the flow of trees is from smaller to larger clusters. Turcotte et al. (1999) quantified this inverse cascade by introducing a cluster

coalescence rate cij, which is the rate at which Ni clusters of a r e a A i coalesce with Nj clusters of area Aj. The rate of coalescence c 0 is proportional to the cluster numbers Ni and Nj and to the linear dimensions of the clusters ri and rj: cq~NiNj~.

(9)

We further hypothesize a power-law (fractal) scaling between the linear dimension of a cluster and its area: ri ~ A~i,

rj ~ A~.

(lO)

For Euclidian clusters, for example a 'circle' of occupied sites, then ~ = 1/2. There are two contributions to the fractal structure of clusters. The first is due to the density of trees in the cluster. If the density of trees is low, then the cluster will have a larger radius relative to the number of trees in the cluster. The second effect is the irregularity of the boundary of the cluster. If the cluster has thin arms (tentacles) that reach out, these will result in an increased rate of coalescence. The studies of Gabrielov et al. (1999) and Yakovlev et al. (2005) have shown that, at any given time, the number of clusters on the grid Ncluste r with area AF is given by Ncluster ~ AF (~+1"5).

(11)

Comparing Eqs. (8) and (11), we obtain = ~:+ 0.5.

(12)

The power-law exponent/3 of the scaling relation between the number of fires NF and AF is ( ~ + 0.5). From Eqs. (1) and (12) we have NF ~ AF/3 AF(~+~

(13)

Up to this point we have been examining the scaling of the frequency-area distributions of model fires and clusters. We now examine the fractal nature of the two-dimensional clusters and fires themselves in the forest-fire model, by associating the values of ~ just discussed with the fractal dimension D of the tree clusters and the model fires. The statistical fractal structure of fires and clusters of area AF are identical. This association is illustrated by the fractal construction given in Figure 4. At first order i = 1 (Fig. 4a) we have a single tree with linear size r~ = 1 and a r e a A1 = 1 tree. At second order i = 2, we add four trees in the adjacent sites, as shown in Figure 4b. The

INVERSE CASCADE AND NATURAL HAZARDS

7

Fig. 4. Deterministic fractal construction illustrating the dependence of the rate of coalescence cij on areas A i and Aj. In (a) we consider a single 'tree' so that linear size rl = 1 and a r e a A 1 ---- 1 tree. In (b) we fill the four adjacent sites with trees so that the linear size is now r2 = 3 and area is A2 = 5 trees. In (c) we repeat this construction using the structure given in (b) so that linear size is now r3 = 9 and area A3 = 25 trees. The fractal dimension of this construction is D = In 5/In 3 = 1.46.

linear size is now r2 = 3 and area A2 = 5 trees. At third order i = 3 we take four of the secondorder structures from Figure 4b, in analogy with the transition from Figures 4a to b, with the result given in Figure 4c. The linear size is now r3 = 9 with areaA3 = 25 trees. From Eq. (10) we have

general result relating ri, the linear dimension of a fractal object, to its area Ai. Substitution of Eq. (18) into Eq. (13) gives

(14)

Thus we have related the power-law exponent /3 of the f r e q u e n c y - a r e a distribution of m o d e l fires to the fractal dimension D of the tree clusters. The fractal structure in Figure 4 emphasizes the irregularity of the boundary of the cluster, which dominates over tree density in contributing to high rates of coalescence. This irregularity is also illustrated in the model fires given in Figure 3. L o w fractal dimensions (D small) lead to large collision rates and fewer large clusters and fires (i.e. small/3) in the model. If the clusters are Euclidian, for example, circles of filled sites, the cluster fractal dimension is D = 2. F r o m Eq. (19) this would give / 3 = 1, that is, N ~ A - - 1 , a critical (i.e. power-law) dependence often associated with 'self-organized criticality' (Turcotte 1999). A very comprehensive numerical study of the same two-dimensional forest-fire model that we have considered in this paper has been given by Grassberger (2002). He considered grid sizes up to Ng = 65,536 x 65,536 ~ 4.295 x 10 9 cells, and firing frequencies as small as f = 1/256,000. Under a wide range of conditions, he found that the f r e q u e n c y - a r e a distribution of model fires was well approximated by Eq. (1), taking /3 = 1.19. F r o m Eq. (19), this corresponds to a cluster fractal

J - r j"

If we take A i = A2 = 5, Aj = Aa = 1, ri = r2 = 3, rj = rl = 1 for our fractal construction, we then have 5#=3

(15)

In 3 - - ~ 0.683. ~ = ln5

(16)

The fractal dimension D of this construction can be defined as (Turcotte 1997) D-

In 5 In (Aj/Ai) - -~ 1.465. In (rj/ri) In 3

(17)

If we compare Eq. (16) to Eq. (17), we have 1 = --. D

(18)

Although this result was derived for the particular fractal construction illustrated in Figure 4, it is a

1 /3 = 0.5 + - ~ . /_)

(19)

8

B.D. MALAMUD & D. L. TURCOTTE

dimension of D ---- 1.45. It is interesting to note that the fractal construction illustrated in Figure 4 has a fractal dimension D = 1.46. In Figure 2 we presented f r e q u e n c y - a r e a statistics for wildfires in two different ecoregions in the USA. Although both data sets showed excellent correlation with the power-law relationship (Eq. 1), the exponents were quite different. One potential explanation for these differences would be a variation in the fractal dimension (if one exists) of the combustible materials in the two regions. In the Mediterranean ecoregion division we have/3 = 1.30, and from Eq. (19) the required 'cluster' fractal dimension is D ---- 1.25. In the Subtropical ecoregion division we have/3 = 1.81, and from Eq. (19) D ---- 0.76. Because this is less then D-----1.0, it does not define a cluster. A twodimensional cluster requires that 1.0 < D < 2.0. From Eq. (19) we therefore require 1.0 < / 3 < 1.5. Because /3 = 1.81 for the subtropical ecoregion, the cascade model cannot explain the power-law scaling of fire occurrences.

Discussion and conclusions In this paper we have considered the f r e q u e n c y area statistics of three natural hazards: earthquakes, landslides, and wildfires. In each case, the data can be well characterized by robust power laws. We have also considered three cellular-automata models. In each case 'avalanches' occur and the frequency-area distributions of the avalanches are well approximated by power laws. In order to advance a potential explanation for the robust power-law behaviour observed in both data and CA models, we consider an inverse-cascade model for metastable cluster coalescence. We find that this cascade model also gives robust powerlaw frequency-area statistics with the powerlaw exponent controlled by the fractal dimension of the clusters. The cascade model requires that the power-law exponent fl be in the range 1.0 < / 3 _< 1.5. Thus the inverse-cascade model provides a satisfactory basis for the behaviour of the forest-fire CA model. Many data sets for real wildfires of f r e q u e n c y area distributions also fall into the required range 1.0 < / 3 < 1.5 (e.g. the Mediterranean ecoregion, Fig. 2b); however, others clearly do not (e.g. the Subtropical ecoregion, Fig. 2a). Of the eighteen ecoregion divisions examined by Malamud et al. (2005), half of them have 1.3 < / 3 < 1.5 and the other half 1.5 < / 3 < 1.8. We conclude that the cascade model does not provide a full explanation for the power-law frequency-area statistics of actual wildfires. This is not surprising, because the spread of actual wildfires is much more complex

than the cellular-automata forest-fire model. Nevertheless, the robust agreement of wildfire data with power-law scaling is striking. The contributions of author D.L.T. were partially supported by NSF Grant No. ATM 0327558 and the contributions of author B.D.M. were partially supported by the European Commission's Project No. 12975 (NEST) 'Extreme events: Causes and consequences (E2-C2)'.

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size-distribution of forest fires. Ecological Model-

ling, 150, 239-254. RICOTTA, C., AVENA, G. & MARCHETTI, M. 1999. The flaming sandpile: self-organized criticality and wildfires. Ecological Modelling, 119, 73-77. RICOTTA, C., ARIANOUTSOU, M. ET AL. 2001. Self-organized criticality of wildfires ecologically revisited. Ecological Modelling, 141, 307-311. RUNDLE, J. B., TURCOTTE, D. L., SHCHERBAKOV,R., KLEIN, W. & SAMMIS, C. 2003. Statistical physics approach to understanding the multiscale dynamics of earthquake fault systems. Reviews of Geophysics, 41, article number 1019, doi: 10.1029/2003RG000135, 5-1-5-30. SARKAR, P. 2000. A brief history of cellular automata. ACM Computing Surveys, 32, 80-107. SCHOLZ, C. H. 1997. Size distributions for large and small earthquakes. Seismological Society of America Bulletin, 87, 1074-1077. SOLE, R. V. & MANRUBIA, S. C. 1995a. Self-similarity in rain forests: Evidence for a critical state. Physical Review E, 51, 6250-6353. SOLE, R. V. & MANRUBIA,S. C. 1995b. Are rainforests self-organized in a critical state? Journal of Theoretical Biology, 173, 31-40. TURCOTTE, D. L. 1994. Fractal theory and the estimation of extreme floods. Journal of Research of

the National Institute of Standards and Technology, 99, 377-389. TURCOTTE, D. L. 1997. Fractals and Chaos in Geology and Geophysics, 2rid edn. Cambridge University Press, Cambridge. TURCOTTE, D. L. 1999. Self-organized criticality. Reports on Progress in Physics, 62, 1377-1429. TURCOTTE, D. L. & GREENE, L. 1993. A scaleinvariant approach to flood-frequency analysis. Stochastic Hydrology and Hydraulics, 7, 33-40. TURCOTTE, D. L. & MALAMUD, B. D. 2002. Earthquakes as a complex system. In: LEE, W. H. K., KANAMORI,H., JENNINGS,P. C. & KISSLINGER, C. (eds) International Handbook of Earthquake & Engineering Seismology. Academic Press, London, 209-227. TURCOTTE, D. L., MALAMUD, B. D., GUZZETTI, F. & REICHENBACH, P. 2006. A general landslide distribution applied to a small inventory in Todi, Italy. In: CELLO, G. & MALAMUD,B. D. (eds) FractalAnalysis for Natural Hazards. Geological Society, London, Special Publications, 261, 105-111. TURCOTTE, D. L., MALAMUD, B. D., MOREIN, G. & NEWMAN, W. I. 1999. An inverse-cascade model for self-organized critical behavior. Physica A, 268, 629-643. YAKOVLEV, G., NEWMAN, W. I., TURCOTTE, D. L. & GABRmLOV, A. 2005. An inverse cascade model for self-organized complexity and natural hazards. Geophysical Journal International, 163, 433 -442.

Typical phases of pre-failure damage in granitic rocks under differential compression XINGLIN LEI 1

Institute of Geology and Geoinformation/ GSJ, National Institute of Advanced Industrial Science and Technology (AIST), Higashi 1-1-1, AIST no. 7, Tsukuba 305-8567, Japan (e-mail: xinglin-lei @aist.go.jp) aAlso at State Key Laboratory of Earthquake Dynamics, Institute of Geology, Chinese Earthquake Administration, Beijing, China Abstract: The evolution of pre-failure damage in brittle rock samples subjected to differential compression has been investigated by means of acoustic emission (AE) records. The experimental results show that the damaging process is characterized by three typical phases of microcracking activity: primary, secondary, and nucleation. The primary phase reflects the initial activity of pre-existing microcracks, and is characterized by an increase, with increasing stress, both in event rate and b value. The secondary phase involves subcritical growth of the microcrack population, revealed by an event rate increase and a dramatic decrease of the b value. The nucleation phase corresponds to initiation and accelerated growth of the ultimate macroscopic fracture along one or more incipient fracture planes. During the nucleation phase, the b value decreases rapidly to the global minimum value around 0.5. The temporal variation of b in every phase clearly correlates with grain size of the test sample, hence indicating that a comparatively larger grain size results in a lower b value. In order to investigate the fracture mechanism of each phase, a damage model was tested by employing the constitutive laws of subcritical crack growth of crack populations with a fractal size distribution.

Abundant experimental evidence shows that macroscopic shear failure in rock does not occur by the growth of a single crack in its own plane. Rather, shear failures are preceded by a complex evolution of some pre-failure damage (Lockner et al. 1991; Lei et al. 2004). Therefore, studies focusing on both fracture dynamics and pre-failure damage are a subject of widespread interest, having relevance for several applications, such as safe design of deep tunnelling (e.g. Diederichs et al. 2004), and natural processes such as volcanism and seismology (e.g. Ponomarev et al. 1997; Diodati et al. 2000). The fracturing dynamics and damage evolution in stressed materials has been extensively studied in the laboratory by a number of methods, including (1) direct observation of samples by scanning electron microscopy (e.g. Zhao 1998) or optical microscopy (Cox & Scholz 1988) operated during or after a fracture test and (2) monitoring of the space distribution of acoustic emission (AE) events caused by microcracking (e.g. Lockner et al. 1991; Lei et al. 1992). The fracturing dynamics and the pre-failure damage can be inferred from AE statistics, as the number of AE events that is proportional to the number of growing cracks, and the AE amplitudes are both proportional to the length of crack growth increments in the rock (e.g. Main et al. 1989, 1993;

Sun et al. 1991). Therefore, the AE technique is applied to the analysis of the microcracking activity inside the sample volume, and it can be performed under confining pressure, which is very important for the simulation of underground conditions. In addition, owing to the mechanical and statistical similarities between AE events and earthquakes, AE in rocks is studied as a model of natural earthquakes (Ponomarev et al. 1997). The disadvantage of the AE technique is that it is insensitive to ductile deformation, which does not produce appreciable AE. Therefore, it is applicable only in brittle regimes. The recently developed high-speed, multichannel waveform recording technology permitted monitoring with high precision of the AE events associated with spontaneously/unstably fracturing processes within stressed samples. It was applied to the analysis of the fracture process of hornblende schist (Lei et al. 2000b), of two granitic rocks of extremely different density with a pre-existing microcrack (Lei et al. 2000a), and mudstone containing quartz veins resembling strong asperities (Lei et al. 2000c). The above studies show that the fracturing of several samples containing faults of widely differing strength exhibit three longterm microcracking phases: primary, secondary, and nucleation (Lei et al. 2004), and that the

From: CELLO,G. & MALAMUD,B. D. (eds) 2006. FractalAnalysisfor Natural Hazards. Geological Society, London, Special Publications, 261, 11-29. 0305-8719/06/$15.00 9 The Geological Society of London 2006.

12

X. LEI

density of pre-existing cracks is a major controlling factor in the fracture and pre-failure damage evolution of rocks (Lei et al. 2000a). Besides crack density, the size distribution of pre-existing cracks is another important factor. Following Tapponier & Brace (1976) we assume that the size distribution of pre-existing cracks is consistent with the grain size distribution in each test sample; in other words, we assume that samples of comparatively larger grain size have a larger number of cracks of larger sizes. In order to check this assumption and identify the typical phases during pre-failure damage, systematic experiments were carried out at different loading rates on granites of different grain size distributions (Lei et al. 2005). One major purpose of a study on pre-failure damage is to consider the possibility of predicting the time of the catastrophic failure event from damage evolution. The ever-increasing interest in such predictions is associated with damage models based either on statistics of critical phenomena or on the experimental knowledge about subcritical crack growth. It is well known that subcritical crack growth under stress can be considered a result of stress-aided corrosion at the crack tip. Owing to this, some fracture laws were proposed based on the experimental results of a single macroscopic extensional crack (e.g. Das & Scholz 1981; Meredith & Atkinson 1983). They were later extended to crack populations whose size distribution is fractal (e.g. Main et al. 1993). These types of models predict not only the event release rate, but also the seismic b value (the exponent of the power-law magnitude-frequency relation) (Lei et al. 2005); in addition, they may explain some precursory anomalies, such as quiescence (Main & Meredith 1991) and b value decrease (Main et al. 1989) associated with large earthquakes. The present paper aims on the one hand to confirm the typical phases of pre-failure damage, and on the other hand, to improve available damage models. For this purpose, AE data obtained

from a series of experiments are here analysed and summarized. Experimental data obtained under a constant stress-rate loading (including some creep tests) on samples of several granitic lithologies were used for summarizing the typical common features of pre-failure damage. Such samples were selected in order to check the roles, for the fracturing behaviour, of grain size, of the pre-existing microcrack density, and of macroscopic structures such as healed joints. The first two sections focus on a review of the experimental and data-processing procedures. Typical experimental results are summarized in the third section; then, an improved model is evaluated. Finally, some plausible physical mechanisms occurring during every damage phase are discussed.

Experimental procedure Rock samples

Table 1 lists the analysed lithologies and their mechanical properties, including grain size, density of pre-existing crack, and the minimum/ maximum P-velocities (Vmin, Vmax) along the axial direction. Because most pre-existing cracks are likely to be closed following stress increase, the difference Vma,,- gmin can be considered as a qualitative measurement of pre-existing microcrack density. Some samples contain one or more veins (healed joints), which evolved during compression into the ultimate fracture plane. The size and spatial distribution of pre-existing microcracks are controlled by the grain size distribution (Tapponier & Brace 1976); hence, on the scale range from microcrack size to sample dimension, rocks of larger grain size generally respond more heterogeneously during the fracturing process. The Westerly granite (WG) is fine-grained and has the smallest grain size. The Inada granite (IG), the Tsukuba granite (TG), the Mayet granite (MG) and the granitic

Table 1. Some mechanical properties of the test samples Rock type

Grain size (mm) range/major

Density of pre-existing cracks

Westerly granite (WG) Oshima granite (OG) Inada granite (IG) Granitic porphyry (GP) Tsukuba granite (TG) Mayet granite (MG) S-C cataclasite (SC) Nojima granite (NG)

<2/< 1 1- 5 / ~ 2 1-10/~7 1-10/~6 1 - 3 0 / ~ 10 1 - 3 0 / ~ 15 1-10/~5 1- 10/~5

High High High Very low High Low Very high High

Axial Vp (km s -1) min/max 4.8/5.6 4.4/5.8 4.2/5.8 5.8/6.0 4.2/5.8 5.2/6.1 4.2/5.6 4.7/5.7

PRE-FAILURE DAMAGE IN GRANITIC ROCKS

13

porphyry (GP) are typical coarse-grained lithologies. The Oshima granite (OG) is classified into intermediate-grained rock, and has a mean grain size greater than WG but smaller than IG. Consistent with the value of Vmax- Vmin, the density of pre-existing cracks is high in WG, OG, IG, and TG, whereas it is relatively low in MG, and GP is almost crack-free (Lei et al. 2000a). The WG, OG, and IG are suited for investigating the role of grain size on fracture behaviour, because they have similar mineralogical content and density of pre-existing microcracks, contrasting with significantly different grain size distribution. The MG and TG samples were selected for checking the role of heterogeneity at large scales either because they contained some large grains or because they contained healed joints. A foliated granitic cataclasite (CG) and its host Nojima granite (NG) were also used, in order to investigate the fracture behaviour of such weak rock in or near a fault zone. The SC and NG samples were collected from drill cores on the Nojima active fault zone at a depth of ~200 m. The Nojima fault, located in southwest Japan, ruptured in the 1995 Kobe earthquake (Lin 2001). The experimental results obtained from the MG samples were preliminarily presented by Jouniaux et al. (2001). In the present study, the hypocentre of every AE event was redetennined after checking every arrival time. Such hard work greatly improved the precision of the hypocentres. The test samples were shaped into cylinders of 125 mm or 100 mm length and 50 mm diameter. All samples were dried under normal room conditions for more than one month and then they were compressed at a constant stress rate or at a constant stress (creep test) at a stress level of ~95% the nominal fracture strength at room temperature (air-conditioned at ~25~ During the deformation, the confining pressure was maintained at a constant level of 40 or 60 MPa within the brittle regime. Under these conditions, the rock samples normally fractured along a compression shear fault oriented at ~30 ~ with respect to the axis of maximum compressive stress.

produced by microcracking events. The signal is pre-amplified by 40 dB before feeding into the high-speed waveform recording system, which has a maximum sampling rate of 40 ns and a dynamic range of 12 bits. Two peak detectors were used to capture the values of the maximum amplitudes, from two artificially selected sensors, after 20 or 40 dB pre-amplifiers. An automatic switching subsystem was designed for sequentially connecting every selected sensor, in total as many as 18, to the pulse generator for velocity measurement. Such a high-speed AE monitoring system can record the maximum amplitude and waveform of the AE signals with no major loss of events, even for AE event rates of the order of several thousand events per second such as are normally observed before a catastrophic failure. The AE hypocentres were determined by using the arrival times of the P wave and the measured Pvelocities during every test. Location errors are generally less than 1 - 2 mm for fine-grained rocks and slightly greater for coarse-grained rocks. During every such test, the trigger threshold for waveform recording is about 10 times larger than the threshold for peak detection (i.e. for the detection of the maximum amplitude of the AE signal). In addition, at least four precise P arrival times are required for hypocentre determination. As a result, the hypocentre data are a subset of magnitude data. Besides AE measurement, eight 16-channel, cross-type strain gauges were mounted on the surface of the test samples for measuring the local strains along the axial and circumferential directions. Stress, strain, and confining pressure were digitized at a resolution of 16 bits and sampling interval of millisecond order. The local volumetric strain (ev) was calculated from the axial (Ca) and circumferential (ec) strains according to the equation e~, = ea + 2Ec. The mean strains of the test sample were estimated by averaging these local measurements.

Experimental apparatus

(1)

Figure 1 shows the functional block diagram of the experimental set-up along with important details of the loading apparatus, AE recorder, and other data acquisition systems. The assembled pressure vessel was placed in a loading frame and high-pressure fluid lines attached for external confining pressure. Hydraulic oil was used for the confining pressure medium. As many as 32 PZTs (piezoelectric transducers, compressional mode, 2 MHz resonant frequency, 5 mm in diameter) were mounted on the sample surface for detecting the AE signals

Data processing The statistical parameters which better characterize the pre-failure damaging behaviours are

(2) (3)

energy release rate and cumulative energy release including event rate and cumulative number; b value in the frequency-magnitude distribution; and fractal dimension of the AE hypocentre.

In the following sections it will be briefly illustrated how the above parameters were estimated.

Normalized time-to-failure Rock fracturing is generally characterized as a fast transient phenomenon showing a high degree of

14

X. LEI

Fig. 1. Block diagram of the experimental apparatus used for rock fracture tests. Up to 32 PZT sensors were mounted on the surface of the test sample (a rock cylinder of 125 mm length and 50 mm diameter). The signal is pre-amplified by 40 dB before feeding into the high-speed waveform recording system, which has a maximum sampling rate of 40 ns and a dynamic range of 12 bits. Two peak detectors were used to capture the values of the maximum amplitudes, from two artificially selected sensors, after 20 or 40 dB pre-amplifiers. An automatic switching subsystem was designed for sequentially connecting every selected sensor, in total 18, to the pulse generator for velocity measurement. Stress, strain, and confining pressure were digitized at a resolution of 16 bits and sampling interval of millisecond order.

nonlinearity before the final failure. Plots of AE data v. time-to-failure in logarithmic scale are therefore helpful for presenting the details of the damaging process. The normalized time-to-failure is defined by: _ tf -- t tf -- to

(1)

where tf is the failure time, to is an artificial starting time that can be set, for example, as the onset time of AE activity or any other chosen time. In the present study, for modelling convenience, to was set as the time of the m a x i m u m b value. These time instants correspond to the phase change from an initial rupture to subcritical crack growth of the crack population. This is discussed in detail in later sections. A E m a g n i t u d e a n d energy release rate

Following the definition of the body-wave earthquake magnitude, the magnitude (M) of an AE event is determined according to the log of the

m a x i m u m amplitude (Vm~x) of the AE signal, as it represents the vibration velocity of the elastic wave (Lei et al. 2003), M oc logVm~, = AaB/20; here AoR is the m a x i m u m amplitude in dB. Such definition results in magnitudes consistent with earthquake magnitudes. However, because the calibration parameter of the PZT sensor is unknown, the magnitudes that were obtained ought to be considered only as relative values. The amplitude threshold for the AE signals was set at 45 dB. Hence the AE magnitude was actually calculated using M = (AdB -- 45)/20. We note that the dynamic range of the system is 100 dB. As a result, every event with m a x i m u m amplitude equal to, or larger than, 100 dB would be recorded as M = 2.75. However, such saturation could be easily corrected by using the continuation time of the AE signal (Lei 2003). It should be noted that the m a x i m u m amplitude was digitized with 11 digits. The distinguishable magnitude interval is therefore 0.25. The AE event rate N = d n / d t gives the simplest measurement of the frequency of microcracking activity. The event rate can be calculated for

PRE-FAILURE DAMAGE IN GRANITIC ROCKS either a fixed number of events (dn) or a fixed time interval (dr). The two methods give similar results but may include different details. The generalized energy release correlates with magnitude:

Ei oc 10 CMi

(2)

where C is a constant. The most important cases are C = 0.75 and 1.5, which correspond to the Benioff strain and the classic energy release, respectively. The energy release rate can be estimated by summing Eq. (2) within a given unit time interval:

E = Z IoCM'"

(3)

i

The cumulative energy release is simply defined by

ZE=

IEdt.

(4)

The cumulative event number ~ N can be considered as a special case of Eq. (4) with C = 0.

The b value a n d recurrence time The Gutenberg-Richter relationship for magnitudefrequency distributions holds both for earthquakes and for AE events in rock samples (Scholz 1968; Liakopovlou-Morris et al. 1994). The cumulative number (Nm) of events with magnitude M or greater is a function of the magnitude (Gutenberg & Richter 1944): log10 Nm : a - bM.

(5)

15

where M denotes the mean magnitude estimation (Shi & Bolt 1982). In the present paper, the b values were sequentially calculated for consecutive groups of 500 to 8000 events (n = 500-8000) with an increment of n/4 events. A larger number of events leads to comparatively smoother curves, and it can be used for investigating the long-term trend. In contrast, a smaller number of events leads to a larger scatter, which reflects the complexity of the damage evolution. In the present paper, bLs and bMLH (or b) are used to denote the leastsquare b value and the maximum likelihood b value, respectively. In most experimental studies, the b value was calculated from the signals recorded by one AE sensor. Therefore, the focal distance from the source should be corrected for the attenuation. Nevertheless, the theoretical study by Weiss (1997) has shown that attenuation has no significant effect on the b value. In the present study, the signals from two sensors, mounted at different positions on the sample surface, were fed to peak detectors for recording the maximum amplitude. It was thus found that two sets of amplitude data result in consistent b values, although there are certain differences in the short-term fluctuations (Fig. 2). On the basis of the a and b values, the probabilistic recurrence time Tr for an event with magnitude equal to, or greater than, a chosen M is estimated by

Tr = A T / 1 0 a-bM.

(8)

Here AT denotes the time length for which the a and b values are derived. In the present paper Tr was

In Eq. (5), a and b are constant; an estimate of the b value can be obtained by using either the leastsquares method or the maximum likelihood method (Aki 1965; Utsu 1965; Bender 1983). For given n events, the latter method gives /9 =

n lOgl0 e ~_,in=l (Mi -- Mc + AM/2) loglo e M - Mc + AM/2

(6)

where Mc is the cutoff magnitude and AM is the difference between successive magnitude units, which is 0.25 for the AE data. The approximate standard error of the b value estimate is & = b/v'-n (Aki 1965), and the confidence limit of the estimation is given by

(Mi - ~'lf)2/(n(n - 1)) (7)

a(b) = 2.30b 2 Y i=1

Fig. 2. Maximum likelihood b values calculated from AE signals obtained from two sensors (#1 - solid black line, #2 - solid grey line) are given as a function of the log of the normalized time-to-failure. The b values were sequentially calculated for consecutive data sets of 1000 events with an increment of 250 events. Also shown (grey rectangle in upper right) are the positions of AE sensors #1 and #2 with respect to the WG sample.

16

X. LEI

calculated for the maximum distinguishable magnitude of 2.75.

Fractal dimension o f hypocentre distribution A multifractal analysis was applied to the hypocentre distribution of AE events in order to examine quantitatively the spatial clustering of pre-failure damage. The multifractal concept is a natural extension of the (mono) fractal concept to the case of heterogeneous fractals. It was observed that the AE hypocentres, particularly in coarsegrained rocks, show heterogeneous fractal characteristics (Lei et al. 1993). In the present study, the generalized correlation-integral (Kurths & Herzel 1987), defined by the following equation, was used:

Cq(r) : -[~.~=l(nj(R
(9)

(q = --oo . . . . . --2, -- 1, O, 1, 2 . . . . . oo) where nj(R < r) is the number of hypocentre pairs separated by a distance equal to, or less than, r, q is an integer, and n is the total number of AE events analysed. If the hypocentre distribution exhibits a power law for any q, Cq(r)OCOq, the hypocentre population can be considered as a multifractal, and Dq defines the fractal dimension that can be determined by the least-squares fit on a log-log plot. In the case of a homogeneous fractal, Oq does not vary with q. However, in the case of a heterogeneous fractal, Dq decreases with increasing q and it is called the spectrum of the fractal dimension. It can be easily proved that Do, D1, and D2 coincide with the information dimension, with the capacity dimension, and with the correlation dimension, respectively. The difference between D2 and Doo is an index representing the degree of heterogeneity of the fractal set. Normally, D20 is an efficiently precise estimation of Doo. Fractal analysis was applied routinely either for every group of 100-200 hypocentres, or for every group of 2000-8000 events, in order to inspect the temporal evolution of the fractal structure and the correlation between fractal dimension and b value. For the later case, that is, with a fixed number of events, as mentioned in the previous section, the hypocentre data are a subset of the magnitude data. The average actual number of hypocentres, which were precisely determined and were thus available for fractal analysis in every group, is normally in the order of several hundreds. The minimum number of hypocentres

required for applying fractal analysis was artificially set as 100 for determining reliable D values. The fractal dimension of every group with a number of hypocentres less than 100 was not determined.

Experimental results Systematic experiments on granitic rock samples were performed at different loading rates, in order to investigate the statistical behaviour of prefailure damage and to set an appropriate damage model. A large number of AE events, from tens to hundreds of thousand, were observed during every test. As an example, Figure 3 shows the AE record of a typical test of IG sample subjected to a constant loading rate at 27.5 MPa min -1. As may be seen, the AE data, particularly the energy release rate and the b value, exhibit some typical phases of pre-failure damage, (see also Lei et al. 2003, 2004) and microcracking activity initiated at a stress level of N35% of the sample's fracture strength. The event rate generally increases, with increasing stress, up to the failure point. Following Lei et al. (2003, 2004), in the following sections, a three-phase model including a primary, a secondary, and a nucleation phase, will be used for describing such behaviour. The AE data obtained from the IG, OG, and WG samples are partly summarized in Figure 4, where it is clearly shown that the three distinct phases are a universal feature during rock fracture in the brittle domain. The common features of each phase are summarized in the following sections.

P r i m a r y phase During this phase, microcracking activity was initiated at stress levels of 30-60% of fracture strength of every sample. Such stress values strongly depend on the density and size distribution of pre-existing cracks, and such distributions are controlled by the grain-governed heterogeneity within the sample. In general, a lithology with comparatively larger mean grain sizes or with higher pre-existing microcrack density exhibits a lower initiation stress and a higher AE activity. During the primary phase, the event rate is low and it increases slightly with the increase of stress or time. The b value increases, with increasing stress, from an initial value of 0.5-1.2 to 1.01.4. The initial and final values also depend on the density and size distribution of the pre-existing microcracks. In the fine-grained WG, the primary phase is not clearly distinguishable, involving only a small number of events, with a high

PRE-FAILURE DAMAGE IN GRANITIC ROCKS

17

Fig. 3. An example of AE data obtained from a typical experiment on an IG sample. (a) Differential stress, average axial/circumferential/volumetric strains v. time. (b) AE energy release rate and event rate calculated consecutively for every 10 s. (c) b Value, recurrence time (Tr) value and fractal dimension (D2) v. time. P, S, and N in (b) denote the primary, secondary and nucleation phases, respectively. The time interval and standard error for all b and Dz data are shown by the horizontal and vertical bars, respectively.

initial b value and a small increase (from ~ 1 . 2 to ,~1.4, Fig. 4e, f). In contrast, the primary phase appears obvious in the coarse-grained IG, being characterized by a large number of AE events with a low initial b value and large increase (from ~0.5 to ~ 1.2, Fig. 4a, b). The intermediategrained OG of mean grain size between W G and IG exhibits a somewhat intermediate behaviour between W G and IG, with an increasing b value from ,-0.7 to ~1.3 (Fig. 4c, d). Such results clearly show that the temporal variation of the b value during the primary phase strongly depends on the grain size distribution of every test sample.

Secondaryphase The typical feature of the secondary phase is a microcracking activity in which the event rate increases, with increasing stress or time, and the b value decreases from its maximum at the end of the primary phase. Average values for the maximum b value in WG, OG, and IG are 1.4, 1.3, and 1.2, respectively (Fig. 4). Such maximum results clearly correlated with the major grain size of the test sample; a comparatively larger grain size results in a lower b value. In the next section, it will be shown that the energy release during the secondary phase appears to be consistent with

18

X. LEI 2 MPa min-~, Pc=40 MPa

27.5 MPa min", Pc--40 MPa

,,~

4 i I 's i ...... i , I . / ~ i i i i I i i i .......!......>"-'r--, .......i........>--r--d ........t'-:l [----+--r--4---{~~---r--,--~---r-/rl 1.41 2.8 3

~

2

-'

--~--U=i--{-

i

1.2[ 2.4

~ ~ - - - ~ - I - - - ~ - |

1.oi 2 . o ~

'-'

~1 o

1.6 '~

0

0.61 1.2

-1

o

4 ..........i ........ i ....... i--~,~-c-)i

.i 3 ~:~2

..... , .......... i ........ i ...... i z-

..........i,-ei,lsi--,-L

~-'~i~-[]~

~:i+~ 1"4 12.8 12.4

.~;

:~S

.~ ,..a 1 ............i--=f.........i............~'............f - i .........4..... i 0

......

:ili

1.0 1 2 . 0 -C-

~ ....

.8 11.6

...........i ........ i...... i..... i..... i ........, -i ....... i - - - ! I -

.6 1 1 . 2 -1 i i i i I I I i I i 4 .......4---~---4 ....... ~ l! ...._' 'i..__](3flJ ~ _........ I I i_i ~ __[...... 1.4 t2.8 ..............~( e ) , - - - , - : ~ - - - L ............ 1 ......... 3

"~2

i

i i ...... , . 7 - - , ~ - y ~ - ~

.........!...........I..............~........ i .... i---+--~I;I - - } - - I ..... f f - + - - i - - - f - + ~ , ~ ~ [ ~

,~1 ~

i / .... i.---LF~--.~-.-i

12.4 .

.

.

.

.

.

.

.

.

.

.

i- +

:tt!0

1400 2800 4200 56007000 Time, s

4_

11.6

0 --A--I--4 ~ A - Q - - t .........~--.-'..........~ - ~ - - 4 - - - . 1

[20

........~

~ - - - 1 ] 140 280 420 560 Time, s

0.6 1 1 . 2 700

Fig. 4. The b value, fractal dimension Dz and event rate N, obtained from tests using the IG, OG, and WG s amples under constant stress-rate loading at either 2 MPa min-1 or 27.5 MPa min-1. 'PIS' and 'S IN' denote the transitions from the primary to the secondary and from the secondary to the nucleation, respectively. The maximum-likelihood b values were sequentially calculated for consecutive groups of 1000 events with a running step of 250 events. The fractal dimensions (D2) were calculated for groups of 8000 events, in which the number of precisely determined hypocentres is more than 100.

a damage model based on the constitutive laws for stress corrosion applied to subcritical crack growth of crack populations. Nucleation phase

The nucleation phase includes the nucleation and accelerated growth of failure-related features associated with the macroscopic fracture of the test sample. It is characterized by a rapidly increasing event rate, and by a rapidly decreasing b value tending to a global m i n i m u m of about 0.5. The AE data were collected using a high-speed acoustic emission waveform recording system and applied to granitic rock samples subjected to confined compression tests; these results are summarized in Figure 5. As m a y be observed, the prefailure damage evolution in these samples essentially consists of a typical three-phase pattern. In the fault zone, high AE activity was observed from the start of the test. In contrast, the AE activity began at ~ 9 5 % of the strength of those samples characterized by a very low density of the

pre-existing microcrack. The event rate in the GP sample of Figure 5f was very low before the final fracture, had a very short secondary phase, and a nucleation phase containing only a few foreshocks. However, more than 1000 aftershocks were recorded after the main fracture. Generally, the waveform recorder would be saturated before the final dynamic faulting. The b value in the MG, NG, and TG samples of Figure 5 a - c , e increased during the primary phase from 1.0-1.2 to ~ 1.4. In all cases, the transition from the primary phase to the secondary phase corresponds to the m a x i m u m b value and it can be easily determined. However, the change from the secondary phase to the nucleation phase appears to depend on the sample's grain size and on the pre-existing microscopic and macroscopic structures. When dealing with intact and h o m o g e n e o u s samples, the nucleation phase appears to be a natural extension of the secondary phase. Conversely, in a sample containing a macroscopic structure (Fig. 5 a - c ) or a very low density of pre-existing cracks (Fig. 5f), the transition occurs abruptly and coincides with the

PRE-FAILURE DAMAGE IN GRANITIC ROCKS 3 MPa min -1, Pc---40 MPa

6 MPa min 4, Pc=60 MPa

4

LeG

3 ~2

19

: 7 . . . . !. . . .

l(d)

.... ! - 1.412.8 1.212.4

t - i ......

1.012.0

1

0.8 I 1.6

0 -1 4 3

0.61 1.2 -MG O o i m e d j - - ~ ~

! I

:T,~ L - ~ . - ~ - - ~ - - ~ i ~ r z

~- 1.412.8 1.212.4

~ 2 ....

....

1.012.0 ~

1 0.8 I 1.6

0

0.61 1.2

4

-TG---J ............. ~ l f ~ ]

3 .....i.... ]

-

-

~

~

[..........i........!---i ............i-.-~(f-)-~---i ...... i----i---.] 1.4 12.8 ]-GP (low crack density)i.......[........F--]I 2,

~2 .... i, - - -, i . - -, - ~ J- - - ~i - - i. ~ x, ~ - ~~ ]m ~ . ~ . , . F--+{ -iL - -~- ~ !- L -,+ - ~~- -!H ~ ,, I "1 0 12.0 1 .....6 MPa m ~ ~ - ~ " r | l [--~--~--7--~F~r .... .71 ~, 0, ~.....

'----"-

-

~

'

" & - - - - g - - - - - L - ~ ~ - - , ~ - - ~

0 ~ - - ] - - - ~ - ~ - ~ 0

600

t-

1200 1800 2400 3000 0 Time, s

~3

N

~5 ]

,.6

0.6 11.2 400

800 1200 1600 2000 Time, s

Fig. 5. The b value, fractal dimension D2 and event rate N, obtained from rock samples under constant stress-rate loading. 'P[S' and 'SIN' denote the transitions from the primary to the secondary and from the secondary to the nucleation phases, respectively. The maximum-likelihood b values were sequentially calculated for consecutive groups of 500 events with a running step of 125 events. The fractal dimensions (D2) were calculated for groups of 200 hypocentres.

minimum fractal dimension. The damaged fault zone rock, in sample SC, shows an abrupt onset of the nucleation phase (Fig. 5d,e). In such cases, a diffusion process of AE hypocentres, with an increasing fractal dimension, was observed during the nucleation phase (Fig. 5b-d). The time duration of the nucleation phase varies in a wide range, from a few seconds to ~ 1 0 0 s, and it depends first on sample lithology and pre-existing structures, and secondly on loading conditions. In general, a comparatively lower loading rate results in a longer nucleation phase, and a homogeneous sample has a comparatively shorter nucleation phase. In samples containing large-scale heterogeneous structures, such as joints or large (cm scale) grains, some short-term, large-amplitude fluctuations can be observed in the b value and in the AE rate.

Damage model Subcritical crack growth may occur under stress as a result of stress-aided corrosion at the crack tip.

The quasi-static rupture velocity V has been found experimentally to be related to the stress intensity factor K by Charles's power law (Charles 1958): V = dc/dt

= Vo(K/Ko) l

(10)

where Vo is an initial/detectable velocity and Ko is the corresponding stress intensity factor, c is the crack length (or half-crack length), and I is referred to as the stress corrosion index and has a typical value of 2 0 - 6 0 for polycrystalline rocks under tensile condition (Das & Scholz 1981). This power law is fundamentally based on experimental observations of mode-I crack growth of a single tensile macrocrack. It can be straightforwardly modified for a crack population having a fractal size distribution by introducing an effective mean value for each parameter (Main et al. 1993). However, the physical bases for such an extension have not been discussed in detail. Hereafter, c, K and V indicate the mean values of crack

20

X. LEI

length, stress intensity factor, and growth velocity, respectively. In a generalized form, the stress intensity factor can be expressed as Ki,n,in = Yov/-c, where Y is a dimensionless geometric constant and o- is the remote applied stress. Under loading conditions of constant stress rate w, o - = O-o+Wt, o-o is the stress at t = 0, and the crack growth velocity can be expressed as

c ~1/2 o- l

dc

\Col

dt

__

( C ~ 1/2

--Vo - \co/

(l+wt)

t,

w = og/O-o (11)

where Co is the mean crack length at t = 0. The solution of the above differential equation under the initial condition c = Co at t = 0, is

(l--2)v~

C=Co(1 --

+wt)l+l--1)) 2/(2-l)

2w(l+

1)Co

, w r 0

(12) C=Co(1

(l--2)v~ t]/ 2/(2-~ ,

w=O.

The mean crack length c increases nonlinearly with time. A failure time can be defined as tf so that c approaches infinity:

tf(w)~%((2w(l_.~_l)c 0 ~-1)~1/(/+1,- 1

)

w~0 (13)

2Co

tf=(l_2)v o,

crack growth of a single tensile crack also exhibits a nonlinear dependence on the stress intensity, showing a form similar to Charles's power law:

N=No(K/Ko) t' .

(16)

Within this context, l' is referred to as the 'effective' stress corrosion index and is found to be equal to n within a few percent in brittle rocks (Main & Meredith 1991). From Eq. (16), the event rate can be represented by N-~(1

-t/tf)2r/2-l(1 +wt) l'.

(17)

This model is roughly representative of the AE data in granitic samples under differential compression (Lei et al. 2005). However, the fit of the results typically shows l ' = 12 and l = 2 5 45. We note that l' is only a half to a third of I. This is most likely because l is excessively high for compression tests due to the occurrence of mixed fracture models. Generally, a shear crack should be more stable than a tensile crack. For modelling purposes, it is better to refer to energy release rather than event rate. Under differential compression, the observable data are the magnitude and the hypocentre of the AE events. Thus, a model ought to be derived for the release rate of energy, instead of the increment of crack length. It was found both observationally and theoretically (Kanamori & Anderson 1975) that there is a power-law scaling relation between the release of elastic energy and the rupture area (S):

w=0. E = 10 kSMi oc$3/2;

Under constant stress (w = 0), Eq. (13) reduces to the same equation given by Das & Scholz (1981). Equation (12) can be simplified as c~co(1

--t/tf(W)) 2/2-1, W ~ 0

C=Co(1--t/tf) 2/2-l, w~-O.

Under constant stress-rate loading Eq. (14) is a good approximation of the exact solution (12). Equation (14) can be further simplified by using the normalized time-to-failure defined in Eq. (1):

C=Co'i2/2-l, 7>0.

hence, the increase rates of fracture area can be estimated from AE data by SAn(t)/SAE(O) =

(14)

(15)

This is again a power law. Meredith & Atkinson (1983) showed experimentally that the event rate N of the microcrack activity during the subcritical

(18)

~(lOkSM,) 2/3

Z 10I"OMi

(19)

where the sums are over all events in an individual period of unit time at the given time t. It should be mentioned that Eq. (19) provides an underestimate, because of increasing energy loss with the increase of the mean crack length from the AE source through the receiver. The major factors associated with such loss include (1) frictional heat, (2) scattering of elastic waves through a damaged volume, and (3) progressive replacement of tensile cracking by shear cracking. The total effect of all these factors is assumed to be a function of the mean

PREIFAILURE DAMAGE IN GRANITIC ROCKS crack length c in terms o f a p o w e r law: SAE

=

S c-m

2cc l-m,

=

m

Such values o f l are significantly lower than the typical value o f 2 5 - 4 5 obtained b y using Eq. (16) for the e v e n t rate, and appear more reasonable. The fine-grained samples have smaller m and larger l values (m = 3.0, l = 16) than coarsegrained samples (m = 3.6, l = 12), whereas westerly granite displays smaller m and 1 (m = 2.8, l = 8 - 1 0 ) . It is o b v i o u s that both m and l are independent o f the loading rates. The first term in Eq. (21), w h i c h governs the nonlinear acceleration o f damage, can be ignored w h e n t << tf. H o w e v e r , it dominates w h e n t approaches tf. It is clear that if a rock sample contains s o m e preexisting m a c r o s c o p i c cracks or joints, w h i c h can potentially serve as the ultimate fracture plane, the overall d e f o r m a t i o n b e h a v i o u r of the material could be g o v e r n e d b y such cracks, following their initiation, as a result o f high nonlinearity in the model. In such cases, the d a m a g e m o d e l is basically

(20)

>_ O .

Therefore, the basic equation for modelling the A E data can be expressed b y

E~(t)/E~(O)

+ wt) ~.

= (1 -t/tf)l+2-am/2-t(1

(21)

Here, /!;1 = ~ I 0 I'0M' indicates the energy release rate evaluated b y the m e a s u r e d A E magnitude. The fit o f Eq. (21) to A E data obtained b y typical experiments is s h o w n in Figure 6. As a general rule, except for the primary phase (t > 1), A E energy release data can be well represented b y the aforem e n t i o n e d model. The best parameters, which match the A E data, are m = 2 . 4 - 3 . 6 and l = 8 - 1 6 , depending on the grain size and lithological type. 2 MPa min-', Pc--40Mpa 516 IG :?,~i i I( )i ! , -~~-.................i ..........x..... ~---~ - -I4 [ 5 ........... , ............... b ~ ~ ~ r ............ t----:-~-':%,," s

-

21

27.5 MPa min-1, Pc=40Mpa

[~(;

i j --!-------r

"--J

i

i

i(b) Ti~ W Z ~ ' ~~,'-I i I 1.51o 'T - 4 ........... t ........... 1

~-------I----

!

,E~

! ;d~"~"i

i

|

.]1.11-2

3 I 4 .......~ 1 6 , / - 1 2 - -

i

I O

i

213 - - - i

I--H .........

L s~,= i

:;'i

i

i

L .......................

I

Ol ,

~

i

i

i(e)i

i

I

I

~

I

i

i +.

i

o6~

I

i ...............1

I

, ~ (

f~!

!

~ ---4

~i

!

)I ..--i

i

........ ! ...... L-4

...... i........

0.7 I-4 0.5 I-5 1.510

1.31-1 '

.............. ! 3 14 ::}:i 213 .

.

.

.

i

.

.

.

m=3.0,1=-16

}]~i[i];[J

.

+.......

112

...........i--~r

..........~

' .b_L....................L.........~.........

i

; i

I

j

j

I

SIN"+

!

213

!i

011l-Tii

[.......!

1

i

j

S,N!

T--

t-4

r~

1.11-2 ~

s

-If-......................

0.9 I-3 "~ ,

i

0.71-4

........ 4 .........

i Sl ~N'~I

0 -1 -2 -3 log Normalized time-to-failure

0.7

0.5 t-5

I T i ~ ...........

Z[ i

i

[

.... ---q ............f.......7........r -7~ ...........i"--i __i , ~ ......~ErL_ _.! 1.31-1

..........~-z----i......... i.............. ~.....

112

i

]

1.510

516 WG .<] ](~)i i i i .......... " . . . . . . . . . . . . . . . . ). . . . . . . . . ? ........... ~. . . . . . . . . . F ....... ~ . . . . . .....r ........ - ~ 8 t r e s s - - # - - ~ - ....L..[% ~.,.=zs, l=}O-~t 314 ..... ! ..... "%-i .......... , ""-i- ....... J.........

!ii?

!

i

k ......... i ...............i ............. -i ................

[ i

i

r~

1.11-2

--f ..... ~............? .......... ............ 0.91-3 ~-

--

......

o)

I

0.91-3 ~

[

o

!

s

s

............L] i

..... 9 i ~ ~ ~ i [=5s . . . . .i . . . .___,>rr_~s 1I Ilii ~i i~ S N, ,

516 ,hA

1.3 I-I

1

i i i sl~i ! 0 -1 -2 -3 log Normalized time-to-failure

0.5 I-5

Fig. 6. Key AE data of three granites (IG, OG, WG) having different grain size distribution, under two loading rates of 2.0 and 27.5 MPa min -1. The time axis corresponds to the normalized time-to-failure on a log scale, for presenting the acceleration of the evolution of the pre-failure damage. The dashed lines indicate the fit of the results of the subcritical crack growth model with the energy release rate (E0. 'PIS' and 'SIN' denote the transitions from the primary to the secondary and from the secondary to the nucleation, respectively.

22

X. LEI

applicable. However, the observed data cannot be represented by a single fit. It should be mentioned that for a small value of loading rate w including creep test (w -- 0), neither m nor l can be determined because, for any given constant F, relation (21) gives similar results for every set of (m, l) satisfying m = [(1 + F ) / + 2 ( 1 F)]/2. However, m and 1 can be determined from AE data obtained from tests with a loading rate efficiently larger than zero. In addition, the relative stress intensity factor can be estimated from the AE release rate as

K = (El (t)/E1 (0)) 1/l+2(1-m). Ko

(22)

Meredith & Atkinson (1983) showed that the seismic b value is in fact negatively correlated with the stress intensity K normalized by the fracture toughness K~. Because K/Ko can be estimated from Eq. (22), the correlation of b and K can be expressed alternatively as

estimated by Eq. (22) in fine-grained and in coarse-grained samples both under fast (27.5 MPa min -1) and low (2 MPa min -1) stress rates. The experimental data basically fit the linear correlation defined by Eq. (23); however, data with b values less than 0.8 are systematically not represented by a linear trend. Such scattered data coincide with the last stage, immediately preceding the catastrophic failure. During this stage, larger and larger events frequently occur. A great number of events are embedded within the tail of the previous event. Moreover, events of amplitude larger than 100dB were clipped and recorded as an event of a relative magnitude of 2.75. As a result, both the b value and K/Ko are likely to be underestimated during the last stage.

Discussion

Fracture mechanism o f the p r i m a r y damage p h a s e

b = bo - a(K/Ko).

(23)

Figure 7 shows the maximum-likelihood b values against the normalized stress intensity factor

i 1.4 ,,~-- #o t

1.2

'*'*~

WG

.........

t

~ .~ dh, 2 MPa min-l

27.5 MPa/min "-

0.8

0.6 ----t----J

~,

i

)

1

1.4--

-

| -

1.2

~

; 1

ii

.......................

"~

1.0

.....

i

-~t 2 MPa min-~!

2 i

" FIe o.6

'2

~

....... **ii

i 3

K/K o

i i

i--' -1

- -

! 4

i.... 5

Fig. 7. For two granitic samples WG and IG, the maximum likelihood b values during the secondary and nucleation phases plotted v. K/Ko estimated from the AE energy release rate. Note that Ko at time to corresponds to the start of the secondary phase, or in other words, the start of the subcritical crack growth.

It is clearly recognized that every damage phase is governed by somewhat different fracture mechanisms and that the statistical behaviour of every typical phase strongly depends on the density and size distribution of pre-existing microcracks, which are controlled by grain size distribution within the test sample. The damage model and its fit to experimental data provided some strong insights into the pre-failure damage process in the brittle domain. Based on experimental results, it appears reasonable to assume that the major mechanism of microcracking in the primary phase is associated with some kind of initial rupture of pre-existing flaws. The primary phase shows behaviours that cannot be represented by the damage model based on subcritical crack growth of crack populations. This supports the hypothesis that microcracking during the primary phase is related to the initial rupture of the pre-existing flaws, rather than to subcritical crack growth. The pre-existing microcracks are probably healed prior to loading, and consequently the initial rupture proceeds more easily by separating the crack walls rather than by crack growth. Because a number of mechanisms can originate local tensile stress, and because the extension strength of every crack is much lower than shear strength, it follows that microcrack opening is expected to be the predominant cracking mode during the primary phase. It can be well understood that the primary phase shows an increasing b value with increasing stress, because larger pre-existing microcracks have a higher probability of rupturing at comparatively lower stress.

PRE-FAILURE DAMAGE IN GRANITIC ROCKS As shown in Figures 4 and 5, the event number during the primary phase decreases significantly. In addition, the fractal dimension of the hypocentres showed no significant stress dependence. Compared to smaller grain sizes, the comparatively larger grain size samples show a lower initial b, a larger maximum b, and a lower D2. As the distribution of pre-existing microcracks is controlled by grain size distribution, it is observed that (1) the larger mean grain sizes correspond to a larger number of large cracks and (2) the event rates, the b values, and the fractal dimensions all appear to be consistent with the hypothesis that the primary phase is associated with the initial activity of pre-existing cracks. Therefore, during the primary phase, the b value and fractal dimension reflect the size distribution and spatial distribution of pre-existing microcracks, respectively. Although both parameters appear to depend on grain size distribution of the test sample, no intrinsic correlation was observed between them.

Fracture mechanism of the secondary and nucleation damage phases The damage model illustrated above represents appropriately the AE energy release rate during the secondary and nucleation phases. This implies that the fracture laws governing the mode-I microcracking during a single extensional growth of a macrocrack, also control the mixed mode of microcracking activity under differential compression. Although the brittle behaviour of rocks under compression is complex, it is clear that on a grain or microcrack scale, a variety of mechanisms give rise to localized tensile stress. The formation and growth of shear faults in a very fine-grained rock under triaxial compression is therefore guided by the development of a process zone encompassing tensile microcracks at the fault tip (Lei et al. 2000b). Under differential compression, many AE events also show a nonquadrantal distribution of the P first motions, hence suggesting that they may record the growth of typical wing cracks, that is, a shear cracks with tensile tails (Lei et al. 2000b). Other possible mechanisms causing local tensile stress are: (1) (2) (3)

contacts between grains; bending of elongated grains; and indentation of sharp-cornered grains into neighbouring ones.

As a result, independently of the remote applied stress, mode-I tensile cracking is likely to be the major mechanism for creating a new crack surface and for contributing to subcritical crack growth. Furthermore this common micro-mechanism provides a possible physical base for justifying

23

why fracture laws derived from extension tests of single macroscopic cracks can also be used for representing the statistical behaviour of mixedmode crack populations. The fact that the proposed model represents very well the AE energy release rate during the secondary and nucleation phases also supports the assumption that the damaging process involves mainly the subcritical growth of pre-existing microcracks. Following increase of the mean crack length, the energy release rate increases nonlinearly, while the b value decreases almost linearly with the normalized mean stress intensity factor, which was estimated by the AE release data. The nucleation phase is characterized by a progressively increasing release rate and by a rapidly decreasing b value, which ultimately reaches a global minimum of 0.5. This is the most interesting and important phase during the catastrophic fracture of rock samples that contain heterogeneous faults. The term 'nucleation' is used here rather than 'tertiary phase' because it better describes the nucleation process of the ultimate, unstable fracture. Once the final fault is initiated at one or at several key locations, which could be the edges of asperities or of previously fractured areas, the faulting process will be governed by the accelerating growth of a few very large cracks, or by the progressive fracture of the major asperities on the fault surface. The proposed damage model, when applied to the nucleation phase, emphasizes that the data fit is not satisfactory, due to the governing behaviour of a few large cracks. Following the increase of the mean crack length of the crack population, interaction between cracks becomes in fact more and more important. This interaction results in fluctuations of both the energy release and b value; consequently, macroscopic heterogeneity, such as pre-existing joints or asperities on the final fracture surface, have a strong influence on the damage evolution of the nucleation phase. As already mentioned, the duration time of the nucleation phase varies in a wide range, from a few seconds to ~ 100 s, depending on the density of the preexisting microcracks as well as the loading rate of stress. For any given lithology containing a fault with unbroken asperities, a creep test at constant stress would show a comparatively longer nucleation phase than that recorded at constant stressrate load (Lei 2003). Summarizing the main findings of the nucleation phase, it may therefore be emphasized that (1) this phase records the quasi-static nucleation of the final fault; (2) the damaging process is associated with the progressive fracture of several unbroken asperities on the fault surface; and (3) the AE events caused by the fracture of individual

24

X. LEI

asperities exhibit similar characteristics to those of natural earthquakes, including foreshock, mainshock, and aftershock events. The foreshocks, which initiated at the edge of the asperity, occur with an event rate that increases according to a power law of the temporal distance to the mainshock, and with a decreasing b value (from ~ 1.1 to ~0.5). One or a few mainshocks are then initiated at the edge of the asperity or at the front of the foreshocks. The aftershock period is characterized by a remarkable increase and subsequent gradual decrease in the b value, and by a decreasing event rate, thus obeying the modified Omori law, which has been well established for earthquakes. The fracture of neighbouring asperities is initiated after a mainshock associated to a specific asperity, presumably due to redistribution of the strain energy. The entire process results therefore in the enhancement of stress concentration around the nearest neighbouring intact asperities. The progressive fracturing of multiple, coupled asperities results in some short-term precursory fluctuations both in the b value and in the event rate. However, it is believed that at constant stress-rate loading conditions, this kind of process could be accelerated dramatically.

its host rock of the same lithology (Fig. 5d, e). On the other hand, samples with low pre-existing crack density also have significantly lower b values than those with high pre-existing crack density (Fig. 5f). Finally, both experiments and numerical simulations show that the b value of AE events is controlled by variations of the internal friction angle, which are induced by variations in confining pressure (Amitrano 2003). In conclusion, the b value can be considered a good parameter for measuring the capability of rocks to release accumulated strain energy. With this aim, possible correlations between the b value and crack interaction, as well as assessment of the dependence of the b value on fracture mechanisms, may represent major items for future studies.

The significance of the b value

(1)

Since the 1960s, b-value variations have been directly related to the local stress conditions (Scholz 1968). In the present study, great efforts have been devoted towards investigating the physical significance of the b value in the magnitudefrequency relation, and it as been shown that, in fact, the state of stress plays the most important role in determining the value of b, which is well represented by a linear relation with the stress intensity factor (Fig. 7). However, the present study also shows that, at a given stress condition, the b values strongly depend also on rock heterogeneity, as this is a major factor governing pre-failure damage and fracture dynamics. However, the concept of heterogeneity is somewhat scale-dependent. With regard to sample size, for example, the greatest grain size can be considered as an index of rock heterogeneity. However, for every microcrack, because its size is generally less than the greatest grain, a sample with a comparatively smaller grain may show greater heterogeneity. In general, available experimental data clearly show that the b value, in every phase, obeys the general relationship: bio < boc < bwo, hence suggesting that a comparatively larger grain size results in a lower b value. Furthermore, experimental data show that b values may vary, depending on stress, between 0.5 and 1.5, and that fault zone rocks have a comparatively smaller b value than

Damage localization and failure nucleation It has been shown that the AE hypocentre distribution shows complicated clustering behaviours during the pre-damage evolution. Nevertheless, as shown in Figure 8, there are several common features that may be of interest: In coarse-grained samples, grain size has the role of characteristic scale, leading to a band-limited fractal or bifractal structure. Large bands (L) and small bands (S) may

-0.4

!

[

Majorduster~size~"~

~,,4 N1 /

-0.8 -1.2

~..~.~7...4--"-1"

I ~.~

~

" ~ -1.6 ~--2.0

i.2

~ -2.4

........

-2.8 -3.2 -3.6

.................T ....

D, ~

/ D,,

D~ ~

/ D~ =

a

I P :1.55(.03)/1.07(.07) ,0.99(.04)/0.32(.02) ,125 S : 1 . 7 0 ( . 0 2 ) / 1 . 0 4 ( . 0 2 ) , 1 . 2 8 ( . 0 3 ) / 0 . 9 5 ( . 0 5 ) ,288

....

N 1 : 1 . 3 8 (.01)/i.09 (. 06) ,1.46(. 07}/0.73 (.02) ,273

0

.4

.8

1.2

1.6

Log~o r, m m

Fig. 8. Generalized correlation integral of the AE hypocentres in coarse-grainedand jointed samples. Note that the AE hypocentre exhibits some heterogeneous fractal and bifractal structures; the fractal dimensions estimated for large bands (L) and small bands (S), at q ----2 and 22 are also shown. The numbers in parenthesis are estimation errors. P and S indicate the primary and secondary phases, respectively. N1 indicates the earlier stage of the nucleation phase. The number n indicates the number of hypocentres used for the calculation of the correlation integral.

2

PRE-FAILURE DAMAGE IN GRANITIC ROCKS

(2)

(3)

likely show different fractal dimensions. However, in fine-grained samples, this kind of characteristic scale could not be found, possibly because it is smaller than the available precision of hypocentre determination. AE hypocentre distribution exhibits multifractal features, with changing values of D2-D22, indicating heterogeneity changes during damage evolution (Fig. 8). Damage localization associated with the growth of a fault is characterized by minimum D values measured at the onset of the nucleation phase (Fig. 4c, Fig. 5a-e).

The final phase of damage evolution is of outstanding interest for short-term prediction of catastrophic fracture development. However, the transition from the secondary to the nucleation phase is strongly affected by the homogeneity of the test sample. In samples containing one or more pre-existing joints, a strong localization was observed along some discontinuities that were ultimately ruptured (Fig. 9; see also Satoh et al. 1996). Clear damage localization was also observed in samples with a few large (cm scale) grains, where the final rupture was mostly controlled by the grain boundaries of some large grains. Samples with high pre-existing crack density, such as the foliated granitic cataclasite from the Nojima fault zone, show less brittle behaviour and a gradual damage localization followed by clear diffusion.

25

This results in a significant decrease and subsequent increase in fractal dimension (Fig. 5d). In the case of intact rocks, loaded at a constant stress rate, during the final stage the microcracking occurred so frequently that it was impossible to distinguish every event from the AE waveforms. Hence, it was impossible to clarify the damage localization on the basis of the AE hypocentre distribution. By using the event rate as a feedback signal to control the loading system, the nucleation phase, which would otherwise have taken only a few seconds, could be extended to several hours duration, and the quasi-static nucleation could be mapped using the AE hypocentres (see also Lockner et al. 1991). In addition, the faulting nucleation could be controlled by using an asymmetrical loading cell (Zang et al. 2000) and the result was a final shear fracture, initiated from some artificially determined point, at a relatively lower stress level and lower AE background. In such cases, the damage localization was clearly established.

Relationship between the b value and the fractal dimension In a fractal fault system, the fractal dimension (Ds) of the distribution of fault length is correlated with the b value of earthquakes: Ds = 2b.

(24)

Fig. 9. AE hypocentres in three coarse-grained granites. AE hypocentres during the pre-nucleation (grey circles) and nucleation (black circles) phases. A clear localization of the damage was observed in two samples only. During the nucleation phase, the AE rate was so high that most events occurred on a very noisy background, and it was difficult to determine a sufficient number of P arrival times for hypocentre determination; as a result, the damage localization could not be identified.

26

X. LEI

This result derives from the interrelationships between the frequency-magnitude (Gutenberg & Richter 1944), moment-magnitude (Aid 1967), and moment-source area (Kanamori & Anderson 1975). It would be interesting also to establish whether a similar correlation exists between (1) the b value and the fractal dimension of the spatial distribution of earthquake hypocentres, or (2) the fractal dimension and the spatial distribution of active faults (see also Cello et al. 2006, this volume). Regarding point (1), the AE data by Lockner et al. (1991) show that a decrease of the b value appears to be correlated, with no time shift, with strain localization, that is, with a decrease of the D value. Regarding point (2), both positive and negative correlations are reported (e.g. Oncel et al. 2001). The present study furnishes some additional information on this critical problem, as it has been observed that, during the primary phase, the b value increased with increasing stress, but the fractal AE hypocentre dimension did not show any systematic variation. The b value and fractal dimension reflect, in fact, the size distribution and spatial distribution of pre-existing microcracks, respectively. In other words, there is no intrinsic correlation between the b value and fractal dimension. However, as already mentioned, in some cases, well-defined damage localization and subsequent damage diffusion were observed while evolving from the secondary to the nucleation phases. The change from decreasing to increasing fractal dimension corresponds to the onset of the nucleation phase. However, during the localization-diffusion process, the b value as a long-term average exhibits a monotonous decrease until the final dynamic fracturing. As a result, a positive correlation between b and D2 was observed during the localization stage, while a negative correlation was observed during the diffusion stage. Figure 10 shows the fractal dimension D2 against the b values obtained from the tests shown in Figures 4 and 5. Predictability of rock failure

The experimental results indicate that, either the accelerated energy release rate or the decreasing b value can be used to predict the final catastrophic event successfully. However, for the natural cases, particularly for the prediction of large earthquakes, the situation appears to be much more difficult. First, because large earthquakes are normally nucleated at depths of between a few km and tens of km, it is impossible to obtain sufficient information from the surface-based seismic monitoring networks. Secondly, most large earthquakes occur along well-developed active faults, and are governed by some kind of mixed mechanism

Fig. 10. The fractal dimension D2 of AE hypocentres v. the b value obtained from the tests shown in Figures 4 and 5. In Some cases, D2 correlates linearly with b as De = 2b § constant (see text for details).

including frictional sliding on the fault surface and rock fracture. In spite of this, some precursory phenomena are known to appear months and even years before large earthquakes and volcanic eruptions occur (e.g. Imoto 1991; Hurukawa 1998; Vinciguerra 1999; Zuniga & Wyss 2001; Nuannin et al. 2005). The proposed three-phase damage model may therefore be profitably used for interpreting the sequence of phases preceding catastrophic failure events, including natural processes such as volcanic eruptions, mining-induced rock bursts, and earthquakes. The decrease of the b value and the onset of the nucleation phase, which generally corresponds to the minimum value of the fractal dimension of hypocentres, are key indicators of the preparatory stage of a catastrophic event. More generally, short-term prediction depends on reliable precursory phenomena. A reliable precursor should follow well-defined empirical laws, and should be related to some plausible physical mechanism. Experimental approaches under wellcontrolled conditions are considered useful for finding possible precursors and their related physical mechanism. To this aim, the present study clearly shows that a comparatively greater heterogeneity results in a longer and more complicated nucleation phase, with a larger number and more types of precursory anomalies. Differently stated, the catastrophic failure event can be better predicted in a comparatively more heterogeneous rock mass. It is expected that the interaction between neighbouring cracks becomes increasingly significant as a result of the progressive increase of both density and length of every crack. In heterogeneous rocks, such as in coarse-grained granites, the interactions are markedly affected by the heterogeneous structure, thus causing fluctuations with large amplitude

PRE-FAILURE DAMAGE IN GRANITIC ROCKS in the damage statistics. Such a strong dependence of pre-failure damage on local structure indicates that the predictability of catastrophic events is also site-dependent. Pre-failure damage evolution likely varies greatly in each case; hence, it is particularly important to resolve, for a given target site, the local geological structure. Systematic studies on the simulation of various geological structures are in fact meaningful for prediction purposes. Future studies need therefore to be focused on establishing quantitative correlations between fluctuations of energy release and the heterogeneous structure of the crust.

Conclusions AE data from granitic rock samples indicate that the pre-failure damage process is characterized by three typical phases of microcracking activity: the primary, secondary, and nucleation phases. It was assessed that the evolution of the three-phase pre-failure damage is a common feature within granitic lithologies having different grain size distributions, as well as macrostructures such as joints. The primary phase reflects the initial opening or ruptures of pre-existing microcracks, and it is characterized by an increase, with increasing stress, both of the event rate and the b value. The secondary phase involves the subcritical growth of the microcrack population, revealed by an increase, with increasing stress, in the energy release rate and a decrease in the b value. The nucleation phase corresponds to the initiation and accelerated growth of the ultimate fracture along either one or several incipient fracture planes. During the nucleation phase, the b value decreases rapidly down to the global minimum value of 0.5. Beside the experimental study, a theoretical analysis was performed, in order to improve the damage model based on the constitutive laws of subcritical crack growth for crack populations whose size distribution is fractal. Instead of modelling the event rate as in some earlier studies, the energy release rate was fitted by using this improved model. The model can represent very well the AE energy release rate in three granites of different grain size distribution, implying that the fracture laws (the same laws as those in mode-I microcracking during a single extensional growth of a macrocrack) actually governed the mixed mode of the microcracking activity under differential compression. The fractal dimension of the AE hypocentres appears also to be consistent with the fracture mechanisms peculiar of every phase. During the primary phase, the fractal dimension depends on grain size and it shows no significant temporary variation.

27

Hence it reflects the spatial distribution of preexisting cracks. Some more or less pronounced decrease of the fractal dimension was observed, being associated with the rapid decrease of the b value. The fractal dimension decreased down to a minimum value around the onset of the nucleation phase. A diffusion process of AE hypocentres was also observed, in some cases, following the onset of the nucleation phase. The progressive development of the fracturing through heterogeneous rocks of large grain size, and through samples that include macroscopically heterogeneous structures such as joints, results in some short-term precursory fluctuations, both in the b value and in the energy release rate. The results of this study indicate that the precursorbased predictability of catastrophic failure is highly dependent on pre-existing heterogeneity and loading conditions. The most important factors are the density and size distributions of pre-existing cracks. The three-phase model proposed in this study appears to be meaningful for transferring the experimental results to real situations associated with both artificial applications and natural processes affecting crustal rocks. This research was supported by the Japan Society for the Promotion of Science under grant 14340128, the Ministry of Science and Technology of China under grant 2004BA601B01 and the National Natural Science Foundation of China under grant 40127002. Helpful reviews by G. P. Gregori and F. Storti are gratefully acknowledged. I would also like to express my gratitude to G. Cello for his helpful comments and suggestions.

References AKI, K. 1965. Maximum likelihood estimate of b in the formula log N = a - bm and its confidence. Bulletin of Earthquake Research Institute, University Tokyo, 43, 237-239. AKI, K. 1967. Scaling law of seismic spectrum. Journal of Geophysical Research, 72, 1217-1231. AMITRANO, D. 2003. Brittle-ductile transition and associated seismicity: Experimental and numerical studies and relationship with the b value. Journal of Geophysical Research, 108(B1), 2044, doi: 10.1029/2001 JB000680. BENDER,B. 1983. Maximum likelihood estimation ofb values for magnitude grouped data. Bulletin of the Seismological Society of America, 73, 831 - 851. CHARLES,R. J. 1958. Static fatigue of glass. Journal of Applied Physics, 29, 1549-1560. Cox, S. J. D. & SCHOLZ,C. H. 1988. Rupture initiation in shear fracture of rocks: an experimental study. Journal of Geophysical Research, 93(B4), 33073320. DAS, S. & SCHOLZ, C. H. 1981. Theory of timedependent rupture in the Earth. Journal of Geophysical Research, 86, 6039-6051.

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DIEDERICHS, M. S., KAISER, P. K. & EBERHARDT, E. 2004. Damage initiation and propagation in hard rock during tunnelling and the influence of nearface stress rotation. International Journal of Rock Mechanics and Mining Sciences, 41, 785- 812. DIODATI, P., BAK, P. & MARCHESONI, F. 2000. Acoustic emission at the Stromboli volcano: scaling laws and seismic activity. Earth and Planetary Science Letters, 182, 253-258. GUTENBERG, B. & RICHTER, C. F. 1944. Frequency of earthquakes in California. Bulletin of the Seismological Society of America, 34, 185-188. HURUKAWA, N. 1998. The 1995 off-Etorofu earthquake: joint relocation of foreshocks, the mainshock, and aftershock and implications for the earthquake nucleation process. Bulletin of the Seismological Society of America, 88, 1112-1126. IMOTO, M. 1991. Changes in the magnitude-frequency b-value prior to large (M _> 6.0) earthquakes in Japan. Tectonophysics, 193, 311-325. JOUNIAUX, L., MASUDA, K., LEI, X.-L., NISHIZAWA, O., KUSUNOSE, K., LIU, L. & MA, W. 2001. Comparison of the microfracture localization in granite between fracturation and slip of a preexisting macroscopic healed joint by acoustic emission measurements. Journal of Geophysical Research, 106(B5), 8687-8698. KANAMORI, H. & ANDERSON, D. L. 1975. Theoretical basis of some empirical relations in seismology.

Bulletin of the Seismological Society of America, 65, 1073-1095. KURTHS, J. & HERZEL, H. 1987. An attractor in a solar time series. Physica, 25D, 165-172. LEI, X.-L. 2003. How do asperities fracture? An experimental study of unbroken asperities. Earth and Planetary Science Letters, 213, 345-357. LEI, X.-L., NISHIZAWA, O., KUSUNOSE, K. & SATOH, T. 1992. Fractal structure of the hypocenter distribution and focal mechanism solutions of AE in two granites of different grain size. Journal of Physics of the Earth, 40, 617-634. LEI, X.-L., NISHIZAWA, 0. & KUSUNOSE, K. 1993. Band-limited heterogeneous fractal structure of earthquakes and acoustic-emission events. International Geophysical Journal, 115, 79-84. LEI, X.-L., KUSUNOSE, K., NISHIZAWA,O., CHO, A. & SATOH, Z. 2000a. On the spatio-temporal distribution of acoustic emission in two granitic rocks under triaxial compression: the role of preexisting cracks. Geophysical Research Letters, 27, 1997-2000. LEI, X.-L., KUSUNOSE, K., RAO, M. V. M. S., NISHIZAWA, 0. 8z SATOH, T. 2000b. Quasi-static fault growth and cracking in homogeneous brittle rock under triaxial compression using acoustic emission monitoring. Journal of Geophysical Research, 105(B3), 6127-6140. LE1, X.-L., NISHIZAWA,O., KUSUNOSE, K., CHO, A. & SATOH, T. 2000c. On the compressive failure of shale samples containing quartz-healed joints using rapid AE monitoring: the role of asperities. Tectonophysics, 328, 329-340. LEI, X.-L., KUSUNOSE, K., SATOH, T. & NISHIZAWA, O. 2003. The hierarchical rupture process of a

fault: an experimental study. Physics of the Earth and Planetary Interior, 137, 213-228. LEI, X.-L., MASUDA, K. ETAL. 2004. Detailed analysis of acoustic emission activity during catastrophic fracture of faults in rock. Journal of Structural Geology, 26, 247-258. LEI, X.-L., SATOH, T., NISHIZAWA, O., KUSUNOSE, K. & RAO, M. V. M. S. 2005. Modelling Damage Creation in Stressed Brittle Rocks by Means of Acoustic Emission. Proceedings of 6th Inter-

national Symposium on Rockbursts and Seismicity in Mines (RsSim6), Perth, Australia, 327-334. LIAKOPOVLOU-MORRIS,F., MAIN, I. G. & CRAWFORD, B. R. 1994. Microseismic properties of a homogeneous sandstone during fault nucleation and frictional sliding. International Geophysical Journal, 119, 219-230. LIN, A. 2001. S-C fabrics developed in cataclasitic rocks from the Nojima fault zone, Japan and their implications for tectonic history. Journal of Structural Geology, 23, 1167-1178. LOCKNER, D. A., BYERLEE, J. D., KUKSENKO, V., PONOMAREV, A. & SIDORIN, A. 1991. Quasistatic fault growth and shear fracture energy in granite. Nature, 350, 39-42. MAIN, I. G. & MEREDITH, P. G. 1991. Stress corrosion constitute law as a possible mechanism of intermediate-term and short-term seismic quiescence. International Geophysical Journal, 107, 363-372. MAIN, I. G., MEREDITH, P. G. & JONES, C. 1989. A reinterpretation of the precursory seismic b value anomaly from fracture mechanics. Geophysical Journal, 96, 131-138. MAIN, I. G., SAMMONDS, P. R. & MEREDITH, P. G. 1993. Application of a modified Griffith criterion to the evolution of fractal damage during compressional rock failure. International Geophysical Journal, 115, 367-380. MEREDITH, P. G. & ATKINSON, B. K. 1983. Stress corrosion and acoustic emission during tensile crack propagation in Whin Sill dolerite and other basic rocks. Geophysical Journal of the Royal Astronomical Society, 75, 1-21. NUANNIN, P., KULHANEK, O. • PERSSON, L. 2005. Spatial and temporal b value anomalies preceding the devastating off coast of NW Sumatra earthquake of December 26, 2004. Geophysical Research Letters, 32, L11307, doi:10.1029/ 2005GL022679. ONCEL, A. O., WILSON, T. H. & NISHIZAWA,0. 2001. Size scaling relationship in the active fault networks of Japan and their correlation with Gutenberg-Richter b values. Journal of Geophysical Research, 106(B10), 21827- 21841. PONOMAREV, A. V., ZAVYALOV, A. D., SMIRNOV, V. B. & LOCKNER, D. A. 1997. Physical modelling of the formation and evolution of seismically active fault zones. Tectonophysics, 277, 57-81. SATOH, T., SHIVAKUMAR, K., NISHIZAWA, 0. & KUSUNOSE, K. 1996. Precursory localization and development of microfractures along the ultimate fracture plane in amphibolite under triaxial creep. Geophysical Research Letters, 23, 865-868.

PRE-FAILURE DAMAGE IN GRANITIC ROCKS SCHOLZ, C. H. 1968. The frequency-magnitude relation of microcracking in rock and its relation to earthquakes. Bulletin of the Seismological Society of America, 58, 399-415. SHI, Y. & BOLT, B. A. 1982. The standard error of the magnitude-frequency b value. Bulletin of the Seismological Society of America, 72, 1677-1687. SUN, X., HARDY, H. R. 8,: RAO, M. V. M. S. 1991. Acoustic emission monitoring and analysis procedures utilized during deformation studies on geologic materials. In: SACHSE, W., ROGET, J. 8r YAMAGUCHI, K. (eds) Acoustic Emission: Current Practice and Future Directions. ASTM Special Technical Publication, 365-380. TAPPONIER, P. 8r BRACE, W.F. 1976. Development of stress induced microcracks in Westerly granite.

International Journal of Rock Mechanics and Mining Science Abstracts, 13, 103-112. VINC1GUERRA, S. 1999. Seismic scaling exponents as a tool in detecting stress corrosion crack growth leading to the September-October 1989 flank eruption at Mr. Etna volcano. Geophysical Research Letters, 26, 3689-3692.

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UTSU, T. 1965. A method for determining the value of b in a formula log n = a - b m showing the magnitude-frequency relation for earthquakes (in Japanese). Geophysical Bulletin, 13, 99-103. WHss, J. 1997. The role of attenuation on acoustic emission amplitude distributions and b-values.

Bulletin of the Seismological Society of America, 87, 1362-1367. ZANG, A., WAGNER, F. C., STANCHITS, S., JANSSEN, C. & DRESEN, G. 2000. Fracture process zone in granite. Journal of Geophysical Research, 105(B10), 23651-23661. ZHAO, Y. 1998. Crack pattern evolution and a fractal damage constitutive model for rock. International

Journal of Rock Mechanics and Mining Science, 35, 349-366. ZUNIGA, F. R. 8r WYss, M. 2001. Most- and leastlikely locations of large to great earthquakes along the Pacific coast of Mexico estimated from local recurrence tirne based on b-values. Bulletin of the Seismological Society of America, 91, 1717-1728.

Flow in multiscale fractal fracture networks P. D A V Y 1, O. B O U R 1, J.-R. D E D R E U Z Y 1 & C. D A R C E L 2

1Gdosciences Rennes, UMR 6118 CNRS, Universit~ de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France (e-mail: Philippe.Davy @univ-rennesl.fr) 2ITASCA Consultants SA 64, chemin des Mouilles, 69130 Ecully, France Abstract: The paper aims at defining the flow models, including equivalent permeability, that are

appropriate for multiscale fracture networks. As a prerequisite of the flow analysis, we define the scaling nature of fracture networks that is likely quantified by power-law length distributions whose exponent fixes the contribution of large fractures versus small ones. Despite the absence of any characteristic length scale of the power-law model, the flow structure appears to contain three length scales at the very maximum: the connecting scale, the channelling scale, and the homogenization scale, above which the equivalent permeability tends to a constant value. These scales, including their existence, depend on the fracture length distribution and on the transmissivity distribution per fracture. They are basic in defining the flow properties of fracture networks.

Modelling fluid transfers in fractured rocks still remains one of the main challenges of m o d e m hydrology. Fractures are known to be key structures for the migration of hydrothermal fluids, but their spatial heterogeneity and complexity at all scales make difficult the definition of simplified, but relevant, flow models. A major difficulty arises from the multiple scales involved. Fractures occur at all scales, from microcracks, which may be observed on thin sections, up to plurikilometric faults, which may break the entire crust. Beyond such a characteristic that implies a large distribution of fracture size, the geometry of fracture networks is also characterized by a wide distribution of orientations, of apertures, and by a spatial repartition of fracture densities, which may be inhomogeneous. This complex geometry raises some fundamental issues about the hydraulic characterization and modelling of fractured media. The lack of any apparent characteristic scale for fracture network geometry does not make obvious the definition of a scale that would help to define the basic properties of a relevant fluid flow model. The definition of a representative elementary volume, which is basic to classical homogeneous models, is in particular questionable in mulfiscale heterogeneous systems. Even the definition of a pertinent scale, or scale range, to describe the geometry of a fracture network, or to measure the hydraulic properties of the system, is not explicit. In this paper, we present an insight into a way to deal with multiscale fracture networks. We particularly focus on the consequences of two basic statistical properties of fracture networks: (1) the fracture length distribution and (2) the fracture aperture distribution. Fracture length defines the spatial extent of the heterogeneity that a fracture

may potentially induce on flow; fracture aperture defines the intensity of this flow heterogeneity by fixing fracture transmissivity. We demonstrate in the following sections that these two properties fully define the nature both of flow and of the relationship between transport properties and fracture geometry. Other fracture network characteristics, such as fracture orientation distribution, do not fundamentally change these relationships, and can be easily incorporated in the presented theory with some simple adjustments of the basic model parameters. We assume that the fracture length distribution is a power law. Beyond the relevance of this model to natural fracture networks (which will be discussed in the next section), the power-law distribution model has the interesting property of having no characteristic length scale except the endmost limits - the smallest fracture length lmin, and the largest one /max. Studying the relationship between medium structure and fluid flow for such systems thus illustrates the type of difficulties that we may encounter when dealing with multiscale heterogeneities. The issues of the definition of a relevant scale, or scale range, for fluid transport, and of the pertinent flow model are acutely addressed by such a distribution model. Although this case seems to be widely relevant in natural systems, it is seldom studied in the literature (for discussion of long-range correlated percolation in self-affine surfaces see, however, Weinrib 1984; Prakash et al. 1992; Schmittbuhl et al. 1993; Sahimi & Mukhopdihyay 1996). In this paper, we only treat systems where the matrix is impervious, which applies to most magmatic rocks. The end-member cases of infinitely long fractures, and of infinitesimally small cracks

From: CELLO, G. & MALAMUD,B. D. (eds) 2006. FractalAnalysisfor Natural Hazards. Geological Society, London, Special Publications, 261, 31--45. 0305-8719/06/$15.00 9 The Geological Society of London 2006.

32

P. DAVY ETAL.

have already been fully characterized. Infinite fractures form parallel paths that can be treated analytically (Snow 1969; Kiraly 1975). The specific case of infinitesimally small fracture networks was extensively studied in the framework of the percolation theory (Stauffer & Aharony 1992). We will show that these two end-member cases are encompassed in the power-law length distribution model, and that a much richer phenomenology appears when covering the range of all possible fracture distributions. In the following, we first assess the validity of the power-law length distribution model for natural fracture systems. We then discuss the connectivity property, and eventually the network equivalent permeability as a function of the fracture network structure.

distributions that describe object properties are power laws, the only functions that do not contain any characteristic length scale. The qualification and quantification of the basic power laws of fracture networks have been widely debated over the last fifteen years (Bonnet et al. 2001). A fracture is defined by two basic scales, its length and the average distance to its neighbours, which defines in turn the fracture density. Power laws apply to both the length distribution and the fracture density. We have proposed that the first-order statistics of fracture networks takes the following form: N(1, L) = 7 L D ( a -

l-a

1) x -~+1 /rain

for 1 E [/n~in,/max]

Multiscale geometry Quantifying the scaling properties of fracture networks is crucial at least in assessing the role of the different fracture families. It is really tempting to neglect unobserved fractures when calculating flow, arguing upon their small size. However, small fractures are also numerous, and the balance between the large number of small structures and the small number of large ones is not trivial. In this paragraph our concern is the quantitative determination of such scaling laws. The fact that fractures exist at all observable scales has been known since geology first existed. The intriguing property is that-similar patterns are observed at very different scales (Tchalenko 1970), emphasizing a kind of scale invariance of fracture networks (Fig. 1). Fractal concepts have shed new theoretical light on such peculiar geometry (Mandelbrot 1982), which emphasizes how deeply scales are linked in the fracturing process (Allbgre et al. 1982; King 1983; Turcotte 1986; Davy et al. 1995). A pure fractal network has no intrinsic length scale, and the statistical

Fig. 1. Two-dimensional fracture maps from the Hornelen basin.

(1)

where N(1, L) dl is the number of fractures having a length between l and l + dl in a system of size L, 7 is the fracture density term (number of fracture centres per fractal unit volume), a the exponent of the power-law length distribution, and D the fractal dimension of fracture barycentres, which fixes the scale-dependence of fracture numbers (Davy et al. 1990; Bour et al. 2002). The exponent a is really a quantitative measure of the balance between small and large fractures, because it fixes the ratio between two fracture families of any length I and l', as (l'/1) a. Validating this model and determining its parameters require a detailed description of fracture maps over a large range of scales, a condition that is generally not fulfilled by statistical analysis performed on single outcrop maps (Bonnet et al. 2001). This difficulty can be overcome by analysing fracture networks mapped at different scales of observation (Heifer & Bevan 1990; Yielding et al. 1992; Scholz et al. 1993; Castaing et al. 1996; Odling 1997). An example of such an analysis is given in Figure 2, where seven two-dimensional fracture maps have been used to determine the underlying length distribution of fractures (Bour et al. 2002). The comparison over scales requires knowing the fractal dimension D, which has been measured independently at about 1.8. Except for lengths below the resolution scales, data can all be mapped onto a single power law characterized by an exponent - a , where a is about 2.75 (Bour et al. 2002). The fractal density term (a in Eq. 1) is equal to about 4.0. The consistency of the power-law statistical model over more than two decades has been tested using different methods (Bour et al. 2002). It was made possible by the existence of the extraordinary data set collected by different imaging techniques, from outcrop mapping to aerial photographs (Odling 1997).

FLOW IN FRACTAL FRACTURE NETWORKS

LO= .zs n(1,L)l o~l

-aaonr~rlarlb,~

1@ 10~ lff~

~ Do ' ~ , ~

0 O0 0 ~ 2 2 9 ~ ~ V~'~'v v ~ , ~ . _ ~ O

+++++-[Y~

10-3

[] 0

L=18m L--55m

A

L--90m

V

L--90m L=180m

~

. 0 r 10-2

33

+

L=360m

X

L=720m

. .xX X X X X > ~

lO~ 10-5 10-~ 10-~

lOo

lO~

lO2

xx

Fracture length, l Fig. 2. Computed length distribution for seven fracture maps of the Hornelen basin (see Fig. 1).

The power-law length distribution model was found to describe a large variety of fracture networks, from small-scale joints (opening fractures) to largescale faults (shear fractures) (Segall & Pollard 1983; Scholz & Cowie 5990; Childs et al. 1990; Villemin & Sunwoo 1987; Davy 1993; Odling 1997), and to fit with fracture growth models (Davy et al. 1990; Cowie et al. 1995). The range of measured power-law exponents is quite large (see the compilation in Bonnet et al. 2001), but it is not possible to discriminate between a statistical origin due to difficulties in achieving a robust statistical analysis, and some physical causes coming from the history of fault growth (Cowie et al. 1995), or from the nature of fracturing (Berkowitz et al. 2000). In the example given in Figure 1, which corresponds to a network of joints (tension fracture), we found that the power-law length exponent a and the fractal dimension D nearly fulfil the self-similar relationship a ~ D + 1 (Bour et al. 2002). In contrast, we obtained a much smaller value of the length exponent a for the San Andreas fault system, with a about 2 (Bour & Davy 1999). Note that the power-law length model is still debated. Against such scaling theory is the evidence that intrinsic length scales exist in the physical system, which should control both processes and produced structures. The boundary conditions of the fracturing process and the layering of most geological formations are such length scales that may be identified in fracture distribution models. This addresses the issue of the validity domain of power-law models in terms of physical conditions and/or of scale ranges. Note also that Eq. (1) is

only a first-order description of fracture networks, which gives the average number of fractures of a given length and at a given scale. It can be easily extended to incorporate other fracture properties such as aperture or orientation. However, it does not describe possible correlations between the different geometrical parameters (Bour & Davy 1999; Darcel et al. 2003a). Whatever the parameters, the power-law model predicts the occurrence of large fractures, much larger than the one deduced from classically used exponential or lognormal distributions. Because the fracture length is also a potential correlation distance for flow, this wide length distribution must entail some important consequences for the connectivity of fracture networks, and thus on the network equivalent permeability.

Connectivity In crystalline rocks where the matrix can be considered impervious (Trimmer et al. 1980; National Research Council 1996), fracture network connectivity is a prerequisite to permeability. Unless fractures are large enough, the connectivity is ensured by the fracture clusters that span the whole system. In systems made up of small independent microcracks, the probability of having a spanning cluster (macroconnectivity) is closely related to the probability of connecting two fractures (microconnectivity). This micro to macro upscaling has been successfully described by the percolation theory (Stauffer & Aharony 1992),

34

P. DAVY ETAL.

which statistically relates the global physical properties (connectivity, permeability, and so on) to a density parameter of the global network, p, called the percolation parameter. There are several interesting concepts in percolation theory that help to understand how to deal with heterogeneity in systems close to threshold. The first concept is the correlation length ~, which is the typical size of the largest cluster in unconnected systems, or of the largest non-connected cluster in connected systems. ~: is really a correlation distance for flow; below the percolation threshold, it is the maximum distance for flow transfer; above it is the scale above which the system can be considered homogeneous. The second concept is the percolation threshold, defined as the point where the system goes from disconnected to connected by adding only one crack. This last added crack is among the 'red links', which are essential to ensure connectivity, because if one of them is removed from a connected cluster, the cluster becomes disconnected. In percolation theory, the percolation threshold is obtained at a fixed value of percolation parameter, p = Pc. As p tends to Pc, the correlation length increases to infinity (in practice, it cannot be larger than the system size) at a rate that depends only on the system dimension ~ ~-, ( p - Pc)-" with v = 4/3 in two dimensions and 0.88 in three dimensions. This divergence of the correlation length at percolation threshold due to the addition of a few cracks emphasizes the dramatic contribution of a few microcracks, making impossible the use of homogenization methods to predict the observed variations around the connectivity threshold. The physical properties are also defined by the geometry of the structures that carry flow (Stauffer & Aharony 1992), mainly the infinite cluster, which macroscopically connects the system, and the backbone, which is the part of the infinite cluster that carries a non-nil flow (Fig. 3). At the percolation threshold, the sample-spanning cluster, the backbone, and the red links are fractal at any scale. In the early 1980s, a great number of studies demonstrated that percolation theory applies to networks of small fractures such as microcracks. A major issue was the definition of the percolation parameter, which depends on fracture orientations, fracture density, and fracture lengths. The critical number of intersections per fracture and the critical density of cracks necessary to be at the percolation threshold have been established as functions of the orientation and length distributions (Robinson 1983). In the 1990s, the role played by a widely scattered fracture length distribution was a major concern (Hestir & Long 1990; Berkowitz 1995; Watanabe & Takahashi 1995).

Fig. 3. An example of crack networks at percolation threshold. The connected cluster is in black and the backbone in bold.

The extension of these studies to power-law length distributions, for which there is a significant number of fractures of length larger than the system size, was made both theoretically and numerically in two and three dimensions (Bour & Davy 1997, 1998; de Dreuzy et al. 2000). Except in Bour & Davy (1998), these results were obtained with fractures having a random orientation. From the results of previous studies, we indeed guess that a fracture orientation distribution modifies only slightly the expression of the percolation parameter, but not the applicability of percolation theory as long as length and orientation are not correlated.

Non-fractal networks

For non-fractal networks, we have first shown from intensive numerical simulations that, despite the wide range in length, there exists a percolation parameter p that fully describes system connectivity, which may be written as

p = N • (e) x Ld

(2)

where lL is the fracture length that lies within the system of size L, N is the total number of elements, d is the Euclidean dimension of the system, and e is a form factor accounting for the fracture shape (equal to 1 in two-dimensional systems and to the eccentricity for ellipses in three-dimensional systems) (Bour & Davy 1997, 1998; de Dreuzy et al. 2000). The use of the included length lL

FLOW IN FRACTAL FRACTURE NETWORKS makes possible the extension to networks with fractures larger than the system size. Even when considering length and eccentricity distributions that cover several orders of magnitude, the percolation parameter at the percolation threshold, Pc, remains about constant with small variations mainly due to the eccentricity distribution. Equation (2) makes possible a quantitative evaluation of the contribution of fracture lengths to connectivity. We showed for two-dimensional networks that this contribution depends critically on the length exponent a defined by Eq. (1) (Bour & Davy 1997, 1998; de Dreuzy et al. 2000). We describe the main rules that may be simply extrapolated to three-dimensional or fractal dimensions (Bour & Davy 1998; de Dreuzy et al. 2000):

(1) If a is very large, that is, larger than 3 in two dimensions or 4 in three dimensions, the connectivity is controlled by fractures that are much smaller than the system size (Fig. 4). The classical length distributions such as lognormal, exponential, and gaussian fall into this category, because they decrease with length much faster than any power law. In such a case, percolation theory fully applies with the same geometrical property and classical rules as the ones obtained on networks made up of fractures having all the same length and much smaller than the system size, provided that the mean fracture length is correctly defined. As a consequence, multiscale fracture

(2)

(3)

35

networks can be advantageously replaced by fracture networks of homogeneous length. For a smaller than 2 in two dimensions and 3 in three dimensions, the fracture network connectivity is controlled by fractures having a size of the order of or larger than the system size. The network is equivalent to a superposition of infinite fractures, with a connectivity rule that is simply the probability of encountering these large fractures. In other words, systems are always connected if they are larger than the average distance between two large fractures. For a between these two limits ([2, 3] in two dimensions and [2, 4] in three dimensions), the network connectivity is ruled by a combination of small and large fractures, the terms 'small' and 'large' being defined by comparison with the system size (Fig. 4).

Although the percolation parameter p describes correctly connectivity properties whenever a > 1, the classical percolation theory framework is fully relevant only for networks with a > D + 1, D being the embedding-space (Euclidian) dimension for non-fractal network (the case of a fractal network will be considered in a later paragraph). A basic assumption of this theory is that fractures are supposed to be much smaller than the system size, at least for very large systems. This assumption is not valid if 1 < a < D + 1, because the number of fractures larger than system size L

Fig. 4. Typology of the connectivity of fracture networks with a power-law length distribution characterized by the exponent a.

36

P. DAVY ET AL.

Fig. 5. System connectivity as a function of fracture density and system size in the cases of percolation theory (a) and of the multiscale fracture networks with a < 3 (b). The zone where networks are connected is shaded in grey. Parts (e) and (d) show the increase in permeability with increasing density and system size.

increases as N ( > L ) ~ L -a+D+l. As a very first consequence, which illustrates the failure of percolation theory, the width of the transition (in terms of percolation parameter variations) from nonconnected to connected networks remains large and does not vanish, even for infinitely large fracture systems. It implies that the percolation parameter at threshold depends not only on fracture density as in classical percolation theory, but also on the system size L (Fig. 5). In other words, for a given fracture system at fixed density, the network connectivity increases with scale so that it always becomes well connected at large scales. The critical scale Le at which networks have a probability of 0.5 to be connected can be analytically calculated as a function of fracture parameters (density, powerlaw exponent a, and so on). In a bilogarithmic density-scale diagram, Le corresponds to a diagonal

straight line whose slope depends on the length exponent a (Fig. 5). For L < Lc, the fracture network is on average not connected, although large fractures may occasionally be encountered in small systems. This peculiar scaling behaviour has some practical and theoretical consequences on the network equivalent permeability that we develop below. Fractal networks

For fractal networks, for which D is lower than the Euclidean dimension, the connectivity is also ruled out by fracture clustering encountered at a different scale. This effect was studied by Darcel et al. (2003b) for networks composed of fractures of constant length, and for more complex networks with both a fractal dimension and a power-law length

FLOW IN FRACTAL FRACTURE NETWORKS distribution. In fact, the dependency of network connectivity on fractal clustering was found to be related to lacunarity, which is the exact counterpart of clustering. For any fractal whose fractal dimension is strictly less than the embedding dimension, there exist zones free of fracture centres within the system, with a largest size in a ratio to the system size if the fractal dimension is less than the embedding dimension. This geometrical property explains the decrease of the density with scale, which is basic in defining the fractal property. It also leads to a decrease of system connectivity with scale in networks made of small cracks. For fractal networks with a power-law length distribution, the occurrence of large fractures may compensate fractal lacunas if there are fractures whose lengths are of the order of the largest fractal lacunae, that is, of the system size. This conditions is fulfilled whatever the system scale if and only if a < D + 1 (Berkowitz et al. 2000; Darcel et al. 2003b). If a > D + 1, large systems are always disconnected, and we found non-trivial scaling of the number of fractures at threshold N c ~ L x, with x > 2 in two dimensions (Darcel et al. 2003b), making the percolation parameter in Eq. (2) irrelevant to describe threshold.

Network equivalent permeability The large variability of geometrical and physical characteristics of fracture networks generates highly heterogeneous flow fields, for which classical modelling frameworks such as homogenization may be irrelevant. Accounting for the heterogeneity may change the basic relations between the local medium properties, that is, the fracture characteristics, and the global medium hydraulic properties. In this spirit, we focus on some key issues both of theoretical interest, to understand the flow processes in fractured media, and practical interest, to design appropriate modelling frameworks for the network equivalent permeability: (1)

(2) (3)

How does connectivity influence permeability? What is the effect of the connection length and more generally of the geometrical network structure on permeability? Is there any scale dependence of permeability as there is a scale dependence of connectivity? What are the characteristic flow structures? Is there any evolution of the relevant flow pattern with scale? Is there any homogenization scale?

We address these issues from the main results obtained by de Dreuzy et al. (2001a, b, c) on twodimensional random synthetic networks with the

37

following assumptions on fracture length and aperture distributions. Fracture

transmissivity

A fracture is an open void or a zone of high permeability, which is characterized by a transmissivity value tf, which is the permeability integral over fracture thickness (or aperture) (Hsieh 1998). For open fractures, this transmissivity is supposed to be related to the average fracture aperture to the power 3, by reference to the cubic law of two parallel planes (Witherspoon et al. 1980; Pyrak-Nolte et al. 1988; Renshaw 1995; Ge 1997; Oron & Berkowitz 1998; Dijk & Berkowitz 1999). The permeability of a unique fracture, whose width and length are much larger than the system size (but whose thickness is small and finite), thus decreases with scale as kf ~ t / L , where t is its transmissivity and L the system size (note that the permeability of a regular grid of infinite fractures has a constant permeability proportional to the transmissivity divided by the fracture spacing). In addition to fracture length distribution (1) and scale, the probability distribution of fracture transmissivity n(tf) is a key input parameter in determining the network equivalent permeability. For open fractures, the local transmissivity distribution may be directly derived from the distribution of the average aperture per fracture. The few systematic measurements on fracture apertures seem to indicate that its distribution is a power law (Wong et al. 1989; Belfield 1994), entailing a power-law distribution of individual fracture transmissivity. However, fractures are not only joints - that is, open fractures - but also faults filled by a granular gouge, whose transmissivity may hardly be determined except from hydraulic tests. Statistical analysis on such tests has, in general, led authors to propose a lognormal distribution for tf (Dverstop & Andersson 1989; Cacas et al. 1990; Tsang et al. 1996), with a lognormal standard deviation b as large as 3 and a geometrical mean kmgm. We thus consider both power-law and lognormal distributions as potentially sound for transmissivity per fracture, with main parameters b and tmg, respectively, the lognormal standard deviation and the geometrical mean, and 6 and tmin, the power-law exponent and the minimal fracture transmissivity. Simultaneous modelling of the widely scattered fracture length and fracture transmissivity distributions also raises the concern of possible correlations between these two parameters. If the longest fracture has the largest permeability, we expect strong flow localization within these large structures, and thus a significant increase in the network permeability. The effect of a fracture

38

P. DAVY ETAL.

length-fracture transmissivity correlation is also discussed in the following paragraphs.

Pertinent permeability models f r o m scaling properties, and flow structure The network equivalent permeability is fully defined by both the network structure and local transmissivities. It is rather trivial to say that this twofold information is out of reach of the best geophysical method, in particular for small-scale structures whose contribution to permeability is an important issue when trying to assess a relevant permeability model. The aim of the following sections is to discuss flow structure and permeability models, assuming that both fracture network geometry and fracture transmissivities belong to generic distribution models: power laws for fracture length distribution, and either power-law or lognormal distribution for fracture transmissivity. The analytical description of permeability as a function of the geometrical and analytical characteristics is to be found explicitly in three previous articles (de Dreuzy et al. 2001a, b, c). In this paper, we discuss the main hydrologic models that arise from the range of admissible parameters for these distributions. We first discuss some basic properties that define general hydraulic models, that is, permeability scaling and flow structure. Then we develop in the next five subparagraphs the main hydraulic models. Note that the network permeability that we calculate is a statistical distribution, which is found to be remarkably well fitted by a lognormal function, whether the transmissivity distribution is lognormal or power-law. The distribution was inferred from extensive numerical simulations on two dimensional networks, and practically from some hundreds of realizations for the largest networks containing up to hundreds of thousands of fracture intersections, to hundreds of thousands of realizations for the smallest networks. Scaling is a key issue when attempting to assess the role of heterogeneities on flow. It reveals two basic features: the dimensionality of flow structure, and how network permeability samples local values. The former entails a decrease of network permeability with scale if the dimension of the structure that supports flow is smaller than space dimension. The permeability of a single fracture for instance scales as L-1, where L is the system size. The latter effect (how permeability samples local values) scales as a result of the well-known statistical trends that the probability of encountering extreme values, large or small, increases with system size. Thus, systems whose flow samples preferentially high- (low-, respectively) permeability

domains are likely to have a positive (negative, respectively) network-permeability scaling. This statistical effect fixes the scaling behaviour of one-, two-, and three-dimensional homogeneous grids with a link-permeability distribution (Neuman 1994). According to the different model parameters, permeability was found to have all the possible scaling behaviour - that is, either decreasing with scale, constant with scale, increasing with a limit, or increasing without a limit. The conditions that make the permeability model belong to one of these four scaling relationships are explicated below. The spatial distribution of flow is also a main characteristic of any flow model. Flow localization, for instance, is an expected consequence of spatial heterogeneity that reveals the capacity of fluid to select the path that minimizes viscous dissipation (i.e. that of largest permeability). In the following we use a simplified quantification of flow localization that amounts to comparing the main flow path (i.e. the path that carries the largest flow) with the others. We qualify a system as 'channelled' or 'distributed' depending on whether the main flow path carries more flow than all the others together, or not. Note than the number of fractures in the main flow path is an important characteristic of the permeability structure. The different hydraulic models that we describe in the following text are defined from the style of permeability scaling (decreasing, constant, or increasing) and of flow structure (extremely channelled or distributed). Note that a given network can belong to several hydraulic models depending on scale. Indeed we expect the flow structure to potentially contain three main length scales that may control permeability scaling: (1)

(2)

(3)

the connection length, that is, the crossover scale at which some networks shift from disconnected to connected; the homogenization scale, at which the network equivalent permeability becomes constant (if it does); the channelling scale, above which the flow structure shifts from extremely channelled to distributed.

Percolation-like networks. Percolation-like networks are encountered in two- or three-dimensional networks when the length distribution is largely dominated by fractures much smaller than system size (Charlaix et al. 1987; Stauffer & Aharony 1992; de Dreuzy et al. 2001a, b, c). For the power-law length distribution, this condition is fulfilled in two dimensions if the length exponent a is larger than 3 (de Dreuzy et al. 2001a, b, c). Indeed the probability of occurrence of a fracture of length larger than l, scaling as l -a + 1, decreases

FLOW IN FRACTAL FRACTURE NETWORKS always faster than 1-2, so that fractures as large as the system size occur with a very low probability. Besides, the fracture length distribution has welldefined mean and standard deviation. Numerical simulations confirm that the fracture length distribution does not have any effect on the type of hydraulic model. Another consequence is that the correlation between fracture length and fracture transmissivity is negligible. There is a characteristic scale, the correlation length ~, below which the network permeability decreases with scale and above which it is constant. Below the correlation length, flow is extremely channelled in the highest permeable path, called the critical path, and the network permeability is determined by its least permeable element (the critical bond). The network permeability is thus determined by a single element, giving a simple method for estimating the network permeability. This method, called the critical path analysis, might be used for the upscaling of other phenomena (Charlaix et al. 1987; Friedman & Seaton 1998). Above the correlation length, permeability reaches a limit and flow becomes homogenously distributed. The correlation length is practically the homogenization length and defines the Representative Elementary Volume (REV). The permeability decrease and the correlation length depend on the transmissivity distribution. For the lognormal and log-uniform fracture transmissivity distributions, the permeability of the critical bond decreases and the correlation length increases when the fracture transmissivity distribution broadens (Charlaix et al. 1987; de Dreuzy et al. 2001b). For a power-law fracturetransmissivity distribution with an exponent - 6 , the less permeable fractures are the most numerous (because its exponent 6 is always considered to be larger than 1) and always control the network permeability. In this case the average permeability is fixed by the smallest fracture transmissivity; the rest of the distribution changes neither the network permeability nor the correlation length. Unique-fracture networks. When a is small enough, that is, smaller than 3 in two dimensions (or 4 in three dimensions), some networks below the percolation threshold are connected by a unique crossing fracture in which flow is completely channelled. Such large fractures occur with a probability proportional to L min(3-a3) according to Eq. (1) (de Dreuzy et al. 2001a) (note that this expression is obtained by considering all fractures in a space much larger than L, because some fractures whose centre is outside the system can cross it). For a smaller than 3 (in two dimensions), this type of connectivity that is ensured by a single fracture thus increases with the system scale L.

39

That is why connectivity increases with scale. Because the permeability of a unique fracture decreases with scale as 1/L times the fracture transmissivity, the equivalent network permeability scales as L min(2-a'~ if there is no scale effect on transmissivity (i.e. in the absence of correlation between fracture length and transmissivity). For a < 2, permeability is constant because the increase of connectivity exactly offsets the decrease of fracture permeability. For 3 < a < 2, permeability decreases as L 2-a. If there is a correlation between fracture length and transmissivity, the permeability scaling also includes the transmissivity scaling. For instance, we can easily calculate this scaling in the case of perfect correlation with a power-law fracturetransmissivity distribution: K ~ L min(2-a +/3,0) where/3 is the exponent of the relationship t ~ l ~ (because of the two power-law distributions /3 is equal to ( a - 1 ) / ( 6 - 1). As /3 is positive, the correlation effect tends to reduce the decrease of permeability with scale, and can even lead to a scale increase. This regime prevails below the connectivity scale, that is, below the scale at which networks shift from disconnected to connected. Networks above threshold, for which permeability tends to a constant. In this case, at the percolation threshold, permeability becomes constant and flow becomes homogenously distributed in the network. This means that the connectivity, channelling, and homogenization scales are all equal. This happens for constant fracture-transmissivity networks, that is, networks in which all fractures have the same permeability (absence of fracture transmissivity distribution). The correlation length for a > 3 and the critical connectivity scale for a < 3 play the same role as homogenization length. A network taken above this homogenization length is made up of a superstructure that looks like a grid whose mesh is itself made up of either a succession of links and blobs for a > 3 or of a multiple-path, multiplesegment structure for a < 3 (Fig. 6). In the case where fracture length is 'broadly' distributed (1 < a < 3), permeability increases with scale from the connectivity length to the channelling scale, at which flow comes to be homogenously distributed in the network and where permeability has almost reached its limit. The channelling scale, at which extreme channelling vanishes, is equivalent to a homogenization length, at which permeability stops increasing. It is not by chance that extreme channelling is linked to the increase in permeability. In fact, flow is extremely channelled in structures of increasing permeability with scale (Fig. 7).

40

P. DAVY ETAL. in their generating parameters and in the amplitude of the permeability increase. (1)

(2)

Fig. 6. (a) Typical fracture network generated with a = 2.5 and with a constant fracture transmissivity. Black fractures are from the superstructure of the backbone (b), whereas grey fractures can be put into the mesh element. Parts (c) and (d) are sketches of the multipath-multisegment structure. When the extreme channelling vanishes, permeability stops increasing. Basically, when it exists, the channelling length is the average distance between two of the most permeable channels of the network. This flow model, characterized by extreme flow channelling and a scale-increase of permeability with a limit, occurs in two types of networks as listed in the following. The network types differ

Networks having a power-law length distribution with 1 < a < 3, a lognormal permeability distribution; and no correlation between fracture transmissivity and length. The range of permeability increase only depends on the standard deviation of the fracture transmissivity distribution b and is equal to exp(b2/2). The fracture length distribution determines the homogenization length. The broader the length distribution, the smaller the homogenization length and the faster the permeability reaches its limiting value. Networks having a power-law length distribution such as 2 < a < 3, a lognormal permeability distribution, and a perfect correlation between fracture length and fracture transmissivity; that is, that the largest fracture is the most permeable one. The equivalent permeability increases with scale up to a limit. The amplitude of the permeability increase is much larger than in the case of the absence of correlation.

Networks above threshold f o r which permeability increases without a limit. In two cases of 'broad' fracture length and permeability distributions, the equivalent permeability increases without a limit. The homogenization length is not defined, although the flow structure may be distributed in several paths. The first case corresponds to a lognormal distribution for fracture transmissivity, a fracture length distribution whose characteristic exponent a lies in the range [1,2], and with a perfect correlation between fracture transmissivity and length (de Dreuzy et al. 2001b). In this case, the equivalent network permeability diverges logarithmically. There is no homogenization length but there is still a channelling length at which extreme channelling vanishes. The second case corresponds to a power-law fracmre-transmissivity distribution, a fracture length distribution with a power-law exponent in the range [1,3]. The equivalent network permeability K increases as a power law with scale such as K ~ L ~" with ~-given by (de Dreuzy et al. 2001c)

~-=

[O/1 - -

0/2

"

(a - 1)] x min (3 - a, 1) 3-1

(3)

where 0/1 and 0/2 are two constants close to 1 and 0.1. In this case, the fracture length distribution changes the permeability scaling. The permeability increase is steeper when a becomes closer to 1, that is, when the system becomes mainly composed of large spanning fractures. Three possible cases have

FLOW IN FRACTAL FRACTURE NETWORKS

41

Fig. 7. Illustration of the permeability increase for a given network generated for a = 2.5 and a lognormal fracture transmissivity distribution not correlated with the fracture length distribution characterized by b = 2. The network is analysed at increasing scales from left to right (L = 50, 100, and 200). The evolution of the flow pattern and more precisely the channelling in more permeable structures leads to an increase in the permeability. been found for the flow structure according to the characteristic exponents a and 8 o f the fracture length and permeability distributions:

(1)

(2)

When a and 8 are in zone III o f Figure 8 (roughly 8 > 2), flow b e c o m e s distributed (3)

Fig. 8. Fracture length and permeability are power-law distributed with a and 6 the characteristic exponent respectively. In zone I, flow is channelled in one path whatever the scale. In zone II, flow is first channelled for length scales smaller than the channelling length and becomes distributed in several different paths above. In zone III, flow becomes distributed above the connection length (connection length = channelling length).

a b o v e the connection length; connectivity and channelling lengths are equal. W h e n a and 8 are in zone II of Figure 8, extreme channelling vanishes at a well-defined length scale larger than the connectivity length. Permeability increases with scale, whereas flow is not focused in a unique channel. W h e n a and 8 are in zone I of Figure 8, e x t r e m e channelling exists at all scales. Neither the channelling nor h o m o g e n i z a t i o n lengths are defined.

Fig. 9. Comparison of the field data (points and vertical bars; from Clauser 1992) with permeability data derived from two models of lognormal fracture transmissivity distribution, one with and the other without correlation (solid and dashed lines respectively). The model without correlation was fitted for a -----2.7 and b = 3.5 (dashed line), whereas the model with correlation was fitted for a ----2.2 and b = 0.8 (full line).

P. DAVY ETAL.

42

Comparison with scale effects observed in the field

permeability and in turn into the relevant model of fracture length and permeability distributions. Although the compilation of Clauser does not respect the site uniqueness, it gives some insights into the application of this study. Data display a variation of network permeability over three orders of

Permeability scalings observed in the field (Clauser 1992; Schulze-Makuch & Cherkauer 1997) give a first insight into the relevant models of equivalent

(a) Flow structure

Connecting scale 6 1 V

Lc

I

Channelling scale

(b) Permeability scale effects Power-law aperture distribution logK-logL Log normal aperture distribution

~

-.._.,

.~ /

homogenization scale~i n !

!

2

3 lOgl0Z

Fig. 10. (a) Sketch of the flow structures with two characteristic length scales: the connecting scale Lc, and the channelling scale; (b) Scaling of the equivalent permeability calculated for three different transmissivity (or aperture) models. The homogenization scale ~ is defined as the scale above which the permeability becomes constant. It does not exist for the power-law transmissivity model represented by small black stars.

FLOW IN FRACTAL FRACTURE NETWORKS magnitude, from the laboratory to the regional scale. There is a regional scale (L > 100 m - 1 km), at which the system permeability becomes constant. These qualitative observations suggest at least a selection of the potential relevant networks models presented here. The models that display the same type of permeability scale effects are the models of lognormal permeability distribution without correlation between fracture length and permeability for a in the interval [1,3], and the models of lognormal permeability distribution with correlation for a in the interval [2,3]. Figure 9 presents fits of Clauser's data by models of lognormal fracture transmissivity distribution with and without correlation. These data also rule out the power-law fracture transmissivity model. We note finally that the permeability increase with scale is an effect of both length and permeability distributions. Other models based on only length or permeability distribution lead to a decreasing or constant permeability. The models proposed here are more generally the only isotropic bidimensional models that produce a permeability increase with scale.

Discussion and conclusions In this paper we discuss the consequences of the multiscale nature of fault networks on flow property. The fact that fractures exists at all scales, from microcracks to pluri-kilometric faults, is a challenging issue for defining average flow properties of fracture networks. We first claim that most of the fundamental effects are due to two basic properties of fracture networks: the fracture-length distribution, which quantifies the scaling nature of fractures, and the transmissivity distribution per fracture. The fracture length distribution is reasonably modelled by a power law whose exponent a quantifies the scaling nature of fracture networks. It has been measured on three-dimensional fracture outcrops with values ranging from 2 to 3. The transmissivity distribution is much less known because of the difficulty in measure transmissivity in hydrogeological sites. Lognormal functions or power laws are classically used as transmissivity distribution models, but this has to be taken as a conjecture. The flow model is highly dependent on the scaling property of fracture networks, and more specifically on the power-law length exponent. The two end-member models are the percolationlike model, where all fractures have a length much smaller than the system size, and the singlefracture model, where the fracture organization considers only a few large fractures cross-cutting the system. In between, the probability of occurrence of large fractures (i.e. larger than the

43

studied system) increases with scale as such systems are always connected above a critical connecting scale that depends on the power-law exponent a and on the fracture density. Permeability, as well as other flow properties, is intimately related to the flow structure, which we can characterize by three main length scales, which depend on the fracture length distribution and on the transmissivity distribution per fracture (Fig. 10): (1)

(2)

(3)

the connection length, which defines the crossover scale at which some networks shift from disconnected to connected; the channelling scale, which corresponds to a transition in the flow structure below which the flow is channelled into one main path, and above which it is distributed into several paths; and the homogenization scale, above which the network equivalent permeability becomes constant.

In some cases of fracture and permeability structure, these characteristic scales can be nil or infinite. For instance, the channelling scale does not exist (or is infinite) for small cracks at the percolation threshold, or for a finite number of infinite fractures. Likewise there are cases where the homogenization scale is infinite (Figs 8 and 10); that is, the permeability tends to increase at larger scales. Figure 10 summarizes these scaling effects on flow structure and on permeability.

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Crustal stress crises and seismic activity in the Italian peninsula investigated by fractal analysis of acoustic emission, soil exhalation and seismic data G A B R I E L E P A P A R O 1, G I O V A N N I P. G R E G O R I 1, M A U R I Z I O P O S C O L I E R I 1, I G I N I O M A R S O N 2, F R A N C E S C O

ANGELUCCI 1 & GIORGIA GLORIOSO 3

lIstituto di Acustica O. M. Corbino (CNR), via Fosso del Cavaliere 100, 00133 Roma, Italy (e-mail: giovanni, gregori@ idac. rm. cnr. it; gabriele.paparo @idac. rm. cnr. it; maurizio.poscolieri @ idac.rm.cnr, it; pangelu @tiscalinet, it) 2Dipartimento di Ingegneria Navale, del Mare e per l'Ambiente, Universitgt di Trieste, via Valerio 10, 34127 Trieste and Istituto Nazionale di Oceanografia e Geofisica Sperimentale (OGS), Borgo Grotta Gigante 42/c, 34010 Sgonico, Trieste, Italy (e-mail: marson @univ. trieste, it; marson @ogs. trieste, it) 3Facoltf di Ingegneria, Dipartimento di Ingegneria Navale, del Mare e per l'Ambiente, Universitgt di Trieste, via Valerio 10, 34127 Trieste, Italy (e-mail: giorgiaglorioso@ libero.it) Abstract: Crustal stress can be monitored by acoustic emissions (AE, ultrasound), which give an indication of whether a physical system is subject to stress, either of tectonic or endogenous origin. AE intensity critically depends on the damping of the signal; however, AE signals are clear indicators of the fatigue state of the crustal structures constituting the AE source. This aspect can be studied by fractal analysis of AE time series; these are, however, not suited for earthquake forecasting, as they only denote a changing state involving large lithospheric volumes. Several case histories from Italy show that an increased high-frequency AE activity (200 kHz) occurs approximately seven to eight months in advance of large earthquakes that affect areas of a few hundred kilometres radius, and an increased low-frequency AE activity (at 25 kHz) is observed several weeks in advance. Low-frequency AE also correlate with soil exhalation (water-well chemistry) and CH4, whereas fractal analysis of AE signals recorded close to a 'future' epicentral area gives a clear indication of the evolution of the system from about two months before the mainshock. This suggests that systematic monitoring of crustal stress variations may be used for assessing the time evolution of seismic activity.

Geodynamic phenomena over regional scales are characterized by slow deformation and crustal stress propagation within areas that are much larger than those eventually struck by a major seismic event. Some perturbation propagates through the crust and it originates some local deformation, and wherever a sufficient amount of potential elastic energy was previously stored by the system, an earthquake is eventually triggered. The observational evidence that supports this interpretation is discussed in this chapter, based on data from three different sites in Italy (Table 1). Acoustic emission (AE) in the ultrasound range (what we call high-frequency or HF AE at 200 kHz or 160 kHz) and low frequency (LF AE at 25 kHz) have proved to be very effective for monitoring phenomena that occur at the submicroscopic scale. These data precede by as much as several months some

extreme occurrences. The temporal resolution of AE signals is very high, and the information is point-like, and premonitory phenomena deal with an area of the order of ~ 5 0 0 km linear size. Some additional information was available from the data collected by monitoring the chemistry of a water spring in central Italy. Such information is useful for correlation with AE inferences. A few other techniques are much more effective in terms of spatial coverage over some given area, although the temporal resolution is unavoidably very low. In this respect, refer to Poscolieri et al. (2006) in this volume.

Tectonic framework The Italian region is characterized by a large gradient of lithospheric thickness (Fig. 1), which is very

From: CELLO,G. & MALAMUD,B. D. (eds) 2006. FractalAnalysisfor Natural Hazards. Geological Society, London, Special Publications, 261, 47-61. 0305-8719/06/$15.00 9 The Geological Society of London 2006.

G. PAPARO ET AL.

48 Table 1. Data from three different events Name

Date

Starting time (GMT)

Latitude of epicentre (N)

Longitude of epicentre (E)

M

Depth (km)

13 04 35

40 ~ 683 N

15 ~ 55 E

4.5

5

43 ~ 01.38 43 ~ 01.78 43 ~ 01.95 43 ~ 00.76 42 ~ 54.13 42 ~ 54.11

12 ~ 52.42 12 ~ 50.09 12 ~ 49.90 12 ~ 49.79 12 ~ 56.10 12 ~ 54.75

5.6 5.9 5.0 5.4 5.1 5.5

7.0 8.0 5.7 7.4 2.6 5.5

Molise (Giuliano), Italy (1996) Potenza

3 Apr

Colfiorito, Italy (1997; only shocks with M > 5.0) Assisi

26 Sep 26 Sep 03 Oct 06 Oct 12 Oct 14 Oct

00 09 08 23 11 15

33 40 55 24 08 23

12.89 26.73 22.02 53.23 36.87 10.61

Molise, Italy (2002) Molise

31 Oct

11 32

41 ~ 76

thin in the Tyrrhenian Sea and m u c h thicker under peninsular Italy, and by a strongly varying heat flow distribution (Figs 2 and 3). The lithospheric thickness is defined in different ways. Figures 1 and 4 refer to seismic definition; that is, the asthenosphere is identified with a layer where the speed of the seismic waves is lowered, being suggestive of a partial melt. An alternative definition is geothermal, where the geotherm is extrapolated downward until

14 ~

5.4

eventually reaching the depth where partial melt occurs. Another method considers the elastic response of the lithosphere to topographic weight. All three methods give different, although mutually consistent, results. In any case, the map of the heat flow (Fig. 2) is indicative of such leading morphologic features. Figures 5 and 6 refer to specific features of central and southern Italy that are of interest for the present study.

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Fig. 1. Thickness of the lithosphere derived from seismic measurements. Circles indicate epicentres of earthquakes with M > 6.5 and are proportional to focal volume. Arrows indicate fault plane solutions. Thick dashed lines schematize the distribution of intermediate and deep earthquakes. (After Panza 1984.)

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Fig. 2. Schematic map of the heat flow (after Cermak & Hurtig 1979). (Figure and captions after Suhadolc et al. 1990; see also Suhadolc & Panza 1989.) The Tyrrhenian seafloor (Fig. 7), when compared to other sea floors, resembles the typical morphology of the Pacific Ocean floor. It has a large density of seamounts, often characterized by active volcanism, and geomagnetic anomalies that

show a typical double-eye pattern, a morphology that is likely to be associated with electrical breeding along mid-ocean ridges (Gregori 1993, 2002). The Aeolian island archipelago (west of Calabria) is an island arc associated with a Benioff zone

50

G. PAPARO E T A L . I

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Fig. 3. Contour plot of the heat flow density (mW m -2) in the Tyrrhenian Sea and surrounding areas. Contour lines are at 10 mW m -2. (After Zito et al. 2003.) (Fig. 1), where earthquakes with very deep hypocentres (up to 600 km) occur. The isotopic character of the basaltic rock of the archipelago is also typical of an ocean island basalt (Crisci et al. 1991;

Fig. 4. Map of the top of the low-velocity channel in the asthenosphere in the European area (Duet al. 1998). (Figure and captions after Panza & Romanelli 2001). This map shows the lithospheric thickness, in km, derived from seismic data. The equivalent thickness, derived from a data handling of heat flow data, can be indirectly inferred from the heat flow map (see Fig. 2; in km), according to the procedure reviewed by Pollack et al. (1993), and utilising planetary scale interpolations using a spherical harmonic expansion up to degree n = 12. According to Pollack et al. (1993) local features, such as the Mediterranean area, suffer from significant uncertainties deriving from the strongly uneven distribution of the available measured points, which require a truncation at degree n = 12, which results in every feature having a large spatial gradient being smoothed and averaged. In this regard, compare Figure 4 with the greater spatial detail given in Figure 1.

Esperanqa e t al. 1992; Esperanqa & Crisci 1995), rather than of an island arc (IA) basalt (e.g. Hawkesworth e t al. 1993). The bathymetric map of the whole Mediterranean shows a very flat plateau west of Sardinia, in contrast to the great density of seamounts across the Tyrrhenian Sea, which is up to ~ 4 0 0 0 m deep, and a gigantic megashear (Fig. 8a, b) apparently crosses the entire Mediterranean, striking west to

I": !o Fig. 5. Major graben features of Miocene Early Pleistocene age in the Apennines, and location of seismic events with M > 6.5 that occurred in the area from the year 1000 to 1980. (Figure and captions after Luongo et al. 1996.)

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Fig. 6. Geographic distribution and energy levels of the present seismic activity. (Figure and captions after Luongo et al. 1996.) east from Gibraltar to (maybe) the North Anatolian Fault Zone (NAFZ). Sicily, meanwhile, is thought to be a prolongation of continental Africa. The Sicily channel is characterized by some very shallow waters (~100-200 m). In 1831, a temporary volcanic island (Ferdinandea, Graham, or Giulia Island, 36 ~ 10' N, 12 ~ 43 t E; see Fig. 9) was born in few days offshore southern Italy. This island then disappeared, eroded by sea waves in about eight months (with a short reappearance in July 1863). Some other volcanic geothermal areas in Italy occur west of the Apennines, along the Tyrrhenian margin. Along the eastem coast of Sicily, and down to the African shelf, a deep scarp (~-,2000 m deep) is found. The entire area is characterized by a conspicuous northward push, and by a presumable counterclockwise rotation of the Italian peninsula. Some models suggest that Calabria was detached from Sardinia and later compressed against the Italian peninsula, leading to its typical curved pattern.

The entire strip west of the Apennines displays several geothermal and/or volcanic areas. In conclusion, the geodynamic setting of the region displays several intriguing features that cannot be linked to simple tectonic models (see Mantovani e t al. 2002, and references therein).

The AE dataset Several AE records were collected over the Italian peninsula. The general rationale and data handling of AE time series, including laboratory experiments, are discussed in Gregori & Paparo (2004); here only a brief description is given. An ideal and perfect elastic body releases no AE, as it simply transforms kinetic into potential energy, and vice versa. Perfect elasticity, however, is a mere abstraction; every object that is subject to some deformation, no matter how closely it fits with such an abstraction, will experience some

52

G. PAPARO E T A L .

Fig. 7. Geomagnetic anomaly map of the Italian area and Tyrrhenian Sea (after Chiappini et al. 2000). Note the 'double-eye' pattern characteristic of every seamount, which is suggestive of an electrical breeding, according to Gregori (1993, 2002). submicroscopic flaws, identified with the breaking of some crystal bonds. This kind of process triggers an AE signal that propagates through the solid body. If an AE detector is located on top of a rocky outcrop, our observational probe is thus composed of the detector plus some underground body

volume of unknown extension. AE can propagate only through solid objects; water is an excellent AE conductor, whereas loose ground causes damping of the AE signal. When we detect an AE signal, we therefore pick up some occurrence of the breaking of crystal bonds within a natural

STRESS CRISES IN THE ITALIAN PENINSULA probe, and the signal propagates through our detector. As far as the tectonic setting is concerned, it may generate significant stress within the rock body associated with microdeformations, hence allowing AE to be released. Thermo-elastic (i.e. daily warming and cooling) effects have some approximately diurnal variation (except for the dependence on seasonal and meteorological conditions, and so on). Tidal phenomena certainly enter into play. However, one should consider the crustal thickness involved. For instance, a huge cathedral or a stadium moves up and down during flow and ebb, and no AE is released by them, as their basement is simply moved up and down. We shall observe the AE signal only as far as our huge natural probe experiences some differential deformation by which different subvolumes of it are differently affected by the tide. For instance, in the case of a volcano, the ebb and flow effect causes a temporal variation of the effectiveness of the lid that 'corks' the endogenous hot fluids, and this shall result into a corresponding tidal modulation of the observed AE. In order to get rid of all such phenomena, a weighted moving

53

average on the AE signal is carried out, by which one can recognize the time-varying entity of the phenomenon on a day-to-day basis, averaging over all effects displaying some approximately diurnal (or shorter) period variation. An important point in AE analysis concerns the temporal evolution of their frequency record. In general, one first monitors very high frequencies, corresponding to small micropore cracking. Such tiny pores will progressively coalesce into larger ones, which will therefore release AE at progressively lower frequencies. As an example, in the sketch of Figure 10, AE signals are first recognized at 200 kHz, and then at 25 kHz. A seismic roar follows, and later on some built-up structures (e.g. house doors, windows) vibrate. Finally, when the frequency band enters the typical seismic range ( ~ 0 . 5 1.0 Hz), buildings resonate and eventually collapse. The fatigue of the mechanical structures that yield and release AE can be investigated upon considering that a chain reaction occurs. In fact, consider that whenever some new stress is applied, some new atomic crystal bond breaks, that is, some new submicroscopic flaw shall preferentially occur close to the

Fig. 8. World Stress Map, Release 2004, by the Academy of Sciences and Humanities, Heidelberg, Geophysical Institute, University of Karlsruhe. The figures represent (a) the Mediterranean and (b) the Italian areas. Symbols are: normal faults (NF), strike-slip faults (SS), thrust faults (TF). Unknown tectonic regime (U). Black lines represent major plate boundaries. Colour figures are available online at www.world-stress-map.org (Heidbach et al. 2004).

54

G. PAPARO ET AL.

Fig. 8. (Continued).

point where some previous flaws already existed. Fractal analysis of the AE recorded within a rock volume results therefore in being very effective for assessing their temporal evolution. In fact, a threedimensional distribution of the prime AE sources will be reflected in a random time series of the AE signal, and its fractal dimension Dt (the subscript t denotes that a time series of data is considered) will be Dt ~ 1. In contrast, a perfect planar distribution of the prime ALE sources will give Dt ,-~ 0. Upon analysing the time sequence of measured Dt it is thus possible to trace the evolution of an unknown mechanical crustal structure that will eventually generate an earthquake. This process predicts a progressive lowering of Dt from ,-~1.0 to ,-~0.0. The

algorithm for computing Dt is summarized in the Appendix. The present study emphasizes the great heuristic potential of such a method, which, compared with other experimental techniques, has the great advantage of providing information about some very early stages in the evolution of a geological system. In other words, we suggest that monitoring the hydrologic regime of a river is important for flood prevention, but so is the monitoring of the evolution, in space and time, of crustal stresses leading to earthquake-related phenomena. Monitoring a seismic event p e r se is like monitoring a flood when it occurs. Notwithstanding the dramatic consequences of a catastrophe, just having a simple

STRESS CRISES IN THE ITALIAN PENINSULA

55

Fig. 9. Locations of the different monitoring sites and earthquakes (Table 1 and text) mentioned in the present study. The triangular zone close to Potenza (near Giuliano) includes three sites (here indicated by the vertices of a triangle): Matera (top right), Marsicovetere (left), and Valsinni (bottom), where the Civil Protection of the Regione Basilicata is implementing a permanent AE and accelerometer network.

56

G. PAPARO E T AL.

(a)

time

description of its occurrence is not enough, scientifically speaking; rather, we need information about the evolution of the system that precedes the occurrence of the catastrophe. In Figure 9 are shown the sites where AE records have been or are being collected in Italy:

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Some additional recently started-up sites, although they are not considered here, are located in the north-eastern Alps, at a site close to Palermo, in Sicily, and in Basilicata (where a three-station array is going to be operated). Bagno di Romagna, in the northern Apennines, is the site where geochemical data relative to a hot water spring were available, although only for a limited time interval. Three case histories are discussed here, dealing with earthquakes with epicentres located (1) at Giuliano (close to Potenza, at ,~ 18 km from the AE recording site), (2) at Orchi, in the epicentral area of the Colfiorito earthquake (the AE recording station was set up right at the epicentre of the event when it had just occurred), and (3) at the Molise earthquake site. Details about these events are given in Table 1.

(a)

I

Raponi site, at Orchi, near Foligno, in Umbria, central Italy; Gran Sasso massif, in central Italy; Giuliano, close to Potenza, in southern Italy; Formisano site, close to the top of Vesuvius; and the island of Stromboli.

"~ 12"= 10"

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420

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Fig. 10. Qualitative diagram showing how the recorded AE signals are first released at comparatively high frequencies, corresponding to the yield of smaller pores (observations on the order of 200 kHz, 50 kHz). The signals are not Dirac 6-function distributions, (a) but lognormal distributions, (b) modulation such as tide effects, for example, on the volcanic edifice of Vesuvius, (c) determines an apparently damped oscillation, just analogously to the aftershocks of an earthquake. For explanation see text. (Figure after Paparo & Gregori 2003.)

2

3

days

I'

~'

Fig. 11. AE record at Giuliano, Italy. (a) Multiparametric precursor (after Cuomo et al. 2000). (b) LF (low-frequency) AE (1996-1997; raw 15 min integrated AE signal) preceding seismic activity (epicentres in Southern Italy on the left side of the plot, those in central Italy that is the Colfiorito earthquake, on the fight side). (e) The fractal dimension LFDt evolves for > 2 months from LFDt = --H '~ 1,0 through LFDt ,'~0.45. The shock occurred with epicentre at ~ 18 km from the AE station. (d) LFDt V. time for a longer time lag. Note an AE paroxysm appears to precede the main shock (weaker when the epicentre is farther away).

STRESS CRISES IN THE ITALIAN PENINSULA

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through the sensor, depending on the temporal evolution of the underground structures. In Figure 12 are shown the AE records at the Raponi site, integrated with hot water geochemical data from the Bagno di Romagna spring. Figures 12a and b show the 24 hours smoothed running average of the AE records. There is a clear indication that some 'crisis' of crustal stress preceded the Molise earthquake (with epicentre ~-,400 km away) by ~-,7-8 months, as detected by HF AE, and by ,-~2 months as detected by LF AE. The Bagno di Romagna spring revealed raised CH4 levels almost coinciding with the LF AE paroxysm. In our opinion this suggests that the CH4 molecule is apparently overly large for moving through the (a)

(c)-0,4] Giuliano (11 Jan- 09 Apr 1996) 4,5 arrow= mainseismicevent

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Mar J~ Oct Jan Apr Jurl Sep "D~ 1996-1997 29 5 3 1 1 30 28 27 Fig. 11. (Continued).

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2002 In Figure 11 are shown the AE records from the Molise (Giuliano) earthquake. The most remarkable information emerging from the data set is the slow trend of the LF fractal dimension LFDt, which changes slowly over about two months, from LFDt r,J 1.0 to LFDt ~ 0.5. We also note that the scatter of the plot was probably due to the time-varying efficiency of the waveguide for the AE transport from sources

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Fig. 12. AE record at the Raponi site (Orchi, Foligno), Italy, and at Baguo di Romagna, Italy. (a) The HF AE paroxysm (the large burst appearing on the left part of the figure) that precedes by ~,,7-8 months the Molise earthquake. (b) An abrupt LF AE paroxysm starts almost exactly 2 months before the Molise earthquake. (e) Trend of eFDt. (d) Chemistry of the hot water spring at Bagno shows a possible correlation with the LF AE paroxysm.

I

58

G. PAPARO ETAL. by a period of comparatively large scatter, denoting that the system suffered some stress build-up. When the scatter stopped, the LF AE premonitory rise (Fig. 12b) occurred (the signal was abruptly amplified by a factor of ~20). The LF AE signal was then followed by a trend displaying much greater scatter compared to that recorded before the abrupt premonitory rise. This scatter never disappeared until the occurrence of the main shock. Finally, in Figure 13, we present a curious feature displayed by Vesuvius. Its AE records are characterized by periods of 'inflation', that is, of steadily increasing endogenous hot fluid pressure (generally lasting several months, although at present this state has been going for a few years), and periods of 'deflation'. 'Inflation' is recognized by LFDt '~ 1.0 and 'deflation' by LFDt < 1.0 (Paparo et al. 2004). Towards the end of 2002, the short-lived and abrupt decrease of LFDt appeared suggestive of the beginning of a period of 'deflation', which in reality did not occur. About two days later, however, the Molise earthquake occurred. It appears unlikely that this was a mere coincidence. Potentially, a stress wave crossed Vesuvius and shook its edifice, thus causing some temporary excess exhalation and 'deflation', while the 'inflation' trend soon restarted when the stress wave passed away. The above discussion has dealt with fractal dimension analyses in the time domain, by which significant precursor phenomena may be recognized within the observed AE time series. It is interesting that a fractal analysis in the space domain seemingly

Fig. 12. (Continued). pores of the size typically associated with HF AE, but it can move through the pore size associated with LF AE. Upon closer inspection of Figure 12, a reasonable interpretation can be made for the HF AE plot (Fig. 12a). The first large burst seems to be the actual aforementioned premonitory signal for the Molise earthquake, which was to occur some ~ 7 - 8 months later. The subsequent trend is suggestive of a possible yearly variation, which was also found in the Kefallinla Island (see Poscolieri et al. 2006, this volume), notwithstanding the relevant difference in tectonic setting. This also suggests that the whole region ought to be subject to crustal stress propagation, possibly depending on astronomical forcing. Additional investigations are in progress, although a much longer data base is required. The large premonitory burst that is potentially 'predictive' of the Molise earthquake was followed

Fig. 13. LFDt at Vesuvius, Italy (no diurnal smoothing applied). LEDt dramatically decreases before strong shocks. The LFDt ~0.75 value recorded at the end of 2002 was followed by no local shock; this behaviour was quite anomalous, when compared with over four years of AE records at this site. Because this ~0.75 value of LFDt was recorded two days before the Molise earthquake, it was interpreted as the result of an anomalous behaviour driven by the shaking of the volcano by a transient stress wave propagating through the crust.

STRESS CRISES IN THE ITALIAN PENINSULA gives relevant indications of the maximum magnitude of an earthquake that eventually strikes a given area. In fact, Cello (1997) and Cello et al. (2001) (see also the contribution by Cello and co-workers in the present volume, and references therein) applied the box counting method to active faults, the source being the standard geological map of Italy and specifically mapped fault zones, in order to measure their spatial fractal dimension Ds. As a result, they found that the maximum magnitude of an earthquake is linearly correlated with Ds. The method seems therefore well suited for envisaging the area that is prone to shocks of a given magnitude, although its temporal resolution is very poor (i.e. on the order of several centuries). In contrast, D t values derived by time-series analysis highlight some intriguing results about the time when the shock might be reasonably expected, even though this method is affected by a very poor spatial definition of the epicentral area (several 100 km at least, according to evidence collected in Italy). The combined use of the geostructural approach of Cello and co-workers with information recorded by arrays of simultaneously operated AE recording sites is therefore likely to provide a much better time and space resolution.

Concluding remarks AE records appear very well suited for monitoring the temporal evolution of the crustal stress that precedes the occurrence of earthquakes: HF AE shows 'crises' that precede the seismic shock by ~ 7 - 8 months, and LF AE 'crises' precede it by ~ 2 months. The space definition of the forthcoming shock, however, is very poor and could be improved, as is our understanding of the propagating stress wave. Both these aspects might be addressed by implementing arrays of AE recording sites. Monitored soil exhalation may also be helpful for comparison with AE results. As far as fractal analysis of AE time series is concemed, our results show that it proved to be very effective in providing information on the evolution of the system for > 2 months before the Giuliano earthquake, which was located at about 18 km from the AE recording site. Several investigations by applying fractal analysis to the time series of AE release from stressed laboratory specimens of different materials were carried out (not reported here), and their inferences were helpful for our seismological applications. It appears likely that additional and much relevant information can be obtained, provided that AE array recorders eventually become available. We had a great advantage for comparing phenomena occurring in different tectonic settings. Additional information about system evolution is expected from some of the investigations in progress; these will in

59

fact augment our database with the results from operating recording sites in the Italian Peninsula, and with the available longer data series from already monitored sites. The authors warmly acknowledge the very kind help of Prof. and Mrs. Raponi for hosting and operating our AE recorders on a rocky outcrop in the cellar of their home, Mr. and Mrs. Formisano for hosting and operating our AE recorders on a rocky outcrop in the garden of their house, and Dr. Giovanni Martinelli, who made the Bagno di Romagna records available to us.

Appendix Computing the fractal dimension of a time series of events by the 'box counting m e t h o d ' (bcm) This elementary and well-known algorithm is reported in every textbook (e.g. Turcotte 1992). Self-similarity is the key logical argument by which the fractal algorithm can be applied to a given actual physical observed and real phenomenon. Differently stated, the fractal algorithm can be effectively applied only, and strictly only, whenever the physical system, for its own intrinsic physical reasons (i.e. this is not a matter of mathematics or of logical abstraction; it is a matter of physics), is such as to respect self-similarity or scaleinvariance. By this some features (whatever they be) do actually reproduce themselves identically on different spatial scales. The domain spanned by such scales is represented by the one given interval of the 'ruler', let us say for ]./~1< ]-s "~ ]-/~2. The fractal algorithm is the specific tool for recognizing this intrinsic order, which reflects some physical features. However, in principle, every two physical systems that display an apparently identical order do not need to be controlled by the same physical mechanism. Operatively, we proceed as follows. (1) (2) (3) (4)

(5) (6) (7)

Consider a point-like process P(t) spanning a total time interval 0 < t < To. Define a time interval of duration/z, and call it a 'ruler'. Cover the entire interval 0 < t < To by several contiguous and non-overlapping rulers. Define a counter, where you add 1 every time that one such ruler contains either one or several data points. Call n(/x) the final sum of such a counter. Plot [log n(/z)] v. [log/z]. This is Richardson's plot. Its tilt H is said to be associated with the fractal dimension Dt of the database (see below).

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(8)

The algorithm applies only as long as /-s < /Z < J-s (9) The definition is that/~l is not smaller than the minimum time interval required by the detector and/or by the data acquisition system for constructing the database P(t). (10) Similarly, /.~ must be less than To, because when this upper limit is approached the algorithm becomes nonsense per se. (11) Therefore, the fractal algorithm strictly holds only within some physical boundaries. For additional details refer, for example, to Turcotte (1992, Eq. (2.1) and (7.12)), where it is shown that the fractal dimension Dt ~ - H . The physical meaning can be simply explained as follows. (1)

Let us consider first (Gregori 1998) a series P(t), which is mathematically (i.e. not physically) defined in terms of events that are perfectly and equally time-delayed with respect to each other. (2) Check that, according to definition, Dt ~ 1. (3) Consider this identical mathematical (i.e. not physical) P(t), and change the time instant of every event by defining it by some ideally perfectly random number. (4) Apply the 'bcm', and, as a matter of definition, check that Dt ~ 1. (5) Suppose that the randomness of the time instants is not perfect, and that every newly defined time instant keeps in some way some memory of the previous ones. (6) It can be shown that one must always find 0 < Dt < 1 (and, note, one will never find O t > 1). (7) The more ordered the time series, the closer will Dt be to 0. (8) In the most extreme case that the entire set of events occurs at one and the same time instant (this is the case of maximum reciprocal memory), check that one formally finds Dt ~ 0.

References CELLO, G. 1997. Fractal analysis of a Quaternary fault array in the central Apennines, Italy. Journal of Structural Geology, 19, 945-953. CELLO, G., TONDI, E., MICARELLI, L. & INVERNIZZI, C. 2001. Fault zone fabrics and geofluid properties as indicators of rock deformation modes. Journal of Geodynamics, 32, 543-565. CERMAK, V. 8z HURTIG, E. 1979. Heat flow map of Europe. In: CERMAK, V. & RYBACH, L. (eds) Terrestrial Heat Flow in Europe. Springer, Berlin, 3-40. CHIAPPINI, M., MELONI, A., BOSCHI, E., FAGGIONI,O., BEVERINI, N., CARMISCIANO, C. & M-ARSON, I.

2000. Shaded relief magnetic anomaly map of Italy and surrounding marine areas. Annali di Geofisica, 43, 983-989. CRISCI, G. M., DE ROSA, R., ESPERAN~A, S., MAZZUOLI, R. & SONNINO, M. 1991. Temporal evolution of a three component system: the island of Lipari (Aeolian arc, southern Italy). Bulletin of Volcanology, 53, 207-221. CUOMO, V., LAPENNA, V., MACCHIATO, M., MARSON, I., PAPARO, G., PATELLA, D. & PISCITELLI, S. 2000. Electrical and acoustic anomalous signals compared with seismicity in a test site of southern Apennines (Italy). Physics and Chemistry of the Earth (A), 25, 255-261. Du, Z. J., MICHELINI,A. & PANZA, G. F. 1998. EurlD: a regionalized 3-D seismological model Europe. Physics of the Earth and Planetary Interiors, 105, 31-62. ESPERANfA, S. & CRISCI, G. M. 1995. The Island of Pantelleria: a case for the development of D M M HIMU isotopic composition in a long-lived extensional setting. Earth and Planetary Science Letters, 136, 167-182. ESPERAN~A, S., CRISCI, G. M., DE ROSA, R. & MAZZUOLI, R. 1992. The role of the crust in the magmatic evolution of the island of Lipari (Aeolian islands, Italy). Contributions to Mineralogy and Petrology, 112, 450-462. GREGORI, G. P. 1993. Geo-electromagnetism and geodynamics: 'corona discharge' from volcanic and geothermal areas. Physics of the Earth and Planetary Interiors, 77, 39-63. GREGORI, G. P. 1998. Natural catastrophes and pointlike processes. Data handling and prevision. Annali di Geofisica, 41, 767-786. GREGORY, G. P. 2002. Galaxy-Sun-Earth relations. The origin of the magnetic field and of the endogenous energy of the Earth. Beitriige zur Geschichte der Geophysik und Kosmischen Physik, Band 3, Heft 3. GREGORI, G. P. & PAPARO, G. 2004. Acoustic emission (AE). A diagnostic tool for environmental sciences and for non destructive tests (with a potential application to gravitational antennas). In: SCHRODER, W. (ed.) Meteorological and Geophysical Fluid Dynamics (A Book to Commemorate the Centenary of the Birth of Hans Ertel). Arbeitkreis Geschichte der Geophysik und Kosmische Physik, Wilfried Schr6der/Science, Bremen, 166-204. HAWKESWORTH, C. J., GALLAGHER, K., HERGT, J. M. & MCDERMOTT, F. 1993. Mantle and slab contributions in arc magmas. Annual Review of Earth and Planetary Sciences, 21, 175-204. HEIDBACH, O., BARTH, A., CONNOLLY,P., FUCHS, K., M~LLER, B., REINECKER,J., SPERNER, B., TINGAY, M., WENZEL, F. 2004. Stress maps in a minute: the 2004 world stress map release. Eos Transactions, 85, 521-529. LUONGO, G., OBRIZZO,P. ETAL. Earthquakes prediction in tectonic active areas using space techniques. Proceedings of the Review Meeting on Seismic Risk in the European Union, Vol. 1, 79-105, May 1996.

STRESS CRISES IN THE ITALIAN PENINSULA MANTOVANI, E., ALBARELLO, D., BABBUCCI, D., TAMBURELLI, C. & VITI, M. 2002. Trencharc-backarc systems in the Mediterranean area: examples of extrusion tectonics. Journal of the Virtual Explorer, 8, 131-147. PANZA, G. F. 1984. The deep structure of the Mediterranean-Alpine region and large shallow earthquakes. Memorie della Societgt Geologica ltaliana, 29, 5-13. PANZA, G. F. & ROMANELLI,F. 2001. Beno Gutenberg contribution to seismic hazard assessment and recent progress in the European-Mediterranean region. Earth Science Review, 55, 165-180. PAPARO, G. & GREGORI, G. P. 2003. Multifrequency acoustic emissions (AE) for monitoring the time evolution of microprocesses within solids. AlP Conference Proceedings, Vol. 657. In: THOMPSON, D. O. & CHIMENTI, D. E. (eds) Review of Progress in Quantitative Nondestructive Evaluation, 22, 1423-1430. PAPARO, G., GREGORI, G. P., TALONI, A. & COPPA, U. 2004. Acoustic emissions (AE) and the energy supply to Vesuvius - 'Inflation' and 'deflation' times. Acta geodaetica, et geophysica hungarica, 40, 471-480. POLLACK, H. N., HURTER, S. J. & JOHNSON, J. R. 1993. Heat flow from the Earth's interior: analysis of

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the global data set. Reviews of Geophysics, 31, 267-280. POSCOLIERI, M., LAGIOS, E. ETAL. 2006. Crustal stress and seismic activity in the Ionian archipelago as inferred by satellite- and ground-based observations, Kefallin]a, Greece. In: CELLO, G. & MALAMUD, B.O. (eds) Fractal Analysis for Natural Hazards. Geological Society, London, Special Publications, 261, 63-78. SUHADOLC, P. & PANZA, G. F. 1989. Physical properties of the lithosphere-asthenosphere system in Europe from geophysical data. In: BORIANI, A., BONAFEDE, M., PICCARDO, G. B. & VAI, G. B. (eds) The Lithosphere in Italy. Advances in Earth Science Research. Proceedings of the Convegni Lincei, ACL-80, Academia dei Lincei Pub., Roma, 15-40. SUHADOLC, P., PANZA, G. F. & Mueller, S. 1990. Physical properties of the lithosphere-asthenosphere system in Europe. Tectonophysics, 176, 123-135. TURCOTTE, D. L. 1992. Fractals and Chaos in Geology and Geophysics. Cambridge University Press, Cambridge. ZITO, G., MONGELLI, F., DE LORENZO, S. & DOGLIONI, C. 2003. Heat flow and geodynamics in the Tyrrhenian Sea. Terra Nova, 15, 425-432.

Crustal stress and seismic activity in the Ionian archipelago as inferred by satellite- and ground-based observations, Kefallinia, Greece M A U R I Z I O P O S C O L I E R I 1, E V A N G E L O S

L A G I O S 2, G I O V A N N I P. G R E G O R I 1,

G A B R I E L E P A P A R O 1, V A S S I L I S A. S A K K A S 2, I S S A A K P A R C H A R I D I S 5, I G I N I O M A R S O N 3, K O N S T A N T I N O S FRANCESCO

S O U K I S 4, E M M A N U E L

ANGELUCCI 1 & SPYRIDOULA

V A S S I L A K I S 4,

VASSILOPOULOU 2

llstituto di Acustica O. M. Corbino (CNR), via Fosso del Cavaliere 100, 00133 Roma, Italy (e-mail: giovanni, gregori @idac. rm. cnr. it; gabriele.paparo @idac. rm. cnr. it; maurizio.poscolieri@ idac. rm. cnr. it; pangelu @tiscalinet.it) 2National and Kapodistrian University of Athens, Geophysics Laboratory, Space Applications in Geosciences, Panepistimiopolis, Ilissia, Athens 157 84, Greece (e-mail: sakkas@ geol.uoa.gr; lagios @geol. uoa.gr; Vassilopoulou @geol. uoa.gr) 3Dipartimento di Ingegneria Navale, del Mare e per l'Ambiente, Universit& di Trieste, via Valerio 10, 34127 Trieste, Italy; Istituto Nazionale di Oceanografia e Geofisica Sperimentale (OGS), Borgo Grotta Gigante 42/c, 34010 Sgonico, Trieste, Italy (e-mail: marson @univ. trieste, it; marson @ogs. trieste, it) 4Department of Geology, University of Athens, Athens, Greece (e-mail: evasilak@geol, uoa.gr; soukis @geol. uoa.gr) 5Harokopio University of Athens, Athens, Greece (e-mail: [email protected]) Abstract: Different observational techniques are compared in order to investigate possible

correlations in seismic activity. The study site is the island of Kefallinia (Greece), where measurements available included (1) DInSAR, DGPS, and DEM data, (2) soil exhalation measured by monitoring Radon (Rn) well content, and (3) acoustic emissions (AE) at high and low frequency (point-like records with high temporal resolution). AE records provide: (1) relative time variation of the applied stress intensity and (2) the state of fatigue of stressed rock volumes, the AE source. Our results indicate that the large spatial scale (poor time resolution) may be considered quite satisfactory, whereas fractal analysis of the AE time series displayed some discrepancies when compared to analogous investigations in the Italian Peninsula. Therefore, some refinement is needed in order to reach more precise interpretations of the relevant information available with this kind of data. However, both sets of observations appear in agreement with each other, although more exhaustive investigations would require a suitable array of point-like AE and Rn (or other) measuring sites, as well as longer data series. The latter are particulary helpful for detailed interpretations of the different occurrences within tectonically complex settings where crustal stress crises are marked by various types of geological phenomena.

Slow deformations and crustal stress propagation characterize geodynamic phenomena over regional space scales much larger than the areas struck by a seismic event. The entire scenario can be depicted in terms of a perturbation propagating through the crust and giving rise to local deformations, and eventually triggering an earthquake wherever a sufficient amount of potential elastic energy has been stored by the system. We attempt to discuss the observational evidence eventually supporting such

interpretation, based on observations carried out in the island of Kefallin~a (Cephalonia) in Western Greece. Different observational techniques are considered. The first is measurement of acoustic emission (AE) in the ultrasound range: high-frequency or HF AE at 160 or 200 kHz and low-frequency or LF AE at 25 kHz. These are very effective for monitoring phenomena that occur on a submicroscopic scale and precede extreme events by

From: CELLO, G. & MALAMUD,B. D. (eds) 2006. FractalAnalysisfor Natural Hazards. Geological Society, London, Special Publications, 261, 63-78. 0305-8719/06/$15.00 9 The Geological Society of London 2006.

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several months. The temporal resolution of AE is very high, and the information is point-like, and premonitory phenomena deal with an area as large as ~106 km 2. A few other techniques are much more effective in terms of spatial coverage over a given area, although the temporal resolution is unavoidably very low. For instance, Differential Interferometric Synthetic Aperture Radar (DInSAR) and Differential Global Position System (DGPS) are the best known standard satellite-based techniques for crustal deformation measurements. Topography is another information source, which is utilized by means of suitable Digital Elevation Model (DEM) database. The aims of the present study are (1) to show that DInSAR and DGPS are capable of measuring some important aspects of pre- and post-seismic crustal deformation; (2) to demonstrate the synergistic integration, and complementary use of two satellitebased techniques, with the AE technique and soil exhalation; and (3) to investigate by means of a suitable analysis of a DEM some complementary geomorphologic information related to the tectonic setting of the island.

Tectonic setting The Hellenic arc-trench system (Fig. la) is a tectonically very active area formed 80 Ma BP (millions of years before present) as an oceanic subduction zone. Eventually the area became a case history of interaction between two continental (African and Eurasian) masses. Kefallinia is one of the Ionian Islands in western Greece and it is located on the NW sector of this narrow zone of convergence. The Ionian basin is still being subducted to the south, under the Aegean domain, whereas, to the north, continental collision occurred between the Apulia microplate and the Hellenic foreland (Sachpazi et al. 2000). Those two domains are linked by a major right-lateral N E - S W trending transform fault (KF), located offshore, west of the island (Fig. lb; Louvari et al. 1999; Sachpazi et al. 2000). Kefallinia is built up mainly by alpine MesozoicCenozoic sedimentary rocks belonging to the external units of the Hellenides (Fig. 2a). During the Neogene, they were part of the Hellenide foldand-thrust belt. On top of them, mainly in the south-southwestern sector of the island, younger Plio-Quaternary sediments lay uncomformably (Underhill 1989). The deformation of the Alpine stage is recorded mainly by N W - S E trending thrusts. These older structures are crosscut by more recent N E - S W trending faults, which in some cases exhibit a significant right-lateral movement. Four major tectonic blocks can be distinguished on the island, based on lithology, on similar

structural features, and on a common evolution during the upper Quaternary: (1) the Erissos peninsula block (northern part of the island); (2) the Paliki peninsula block (western part of the island); (3) the Ainos block (central and eastern part of the island); and (4) the Argostoli block (southwestem part of the island). Each of these major blocks consists of several subordinate units and is flanked by a major thrust fault. Most subordinate units have more or less distinct geological features, on account of their difference in geological evolution during some stage. However, during the Late Quaternary, some of them were unified and compressed into the four major blocks of the island (see Fig. 2b). Several strong earthquakes (M > 6.0) have occurred in the Kefallinia area. The last great event (Mw = 6.7) occurred on 17 January 1983, if we do not consider the 14 August 2003 earthquake (Mw----6.2), involving mostly Lefkada island, northeast of Kefallinia (Papadopoulos et al. 2003; Pavlides et al. 2004). Since then, no other event of comparable magnitude has occurred in the area, although a great number of smaller events (4.0 < Mw < 5.0) have been observed, and at least one event of Mw-~ 5.0 is expected every year. The island is considered a Very High Seismic Risk Zone and is the subject of several investigations. We note that there is a substantial difference between the tectonic setting of Kefallinia and that of the Italian peninsula; hence, it is difficult to compare these two geodynamic scenarios.

GPS techniques The DGPS observations were taken from 23 GPS stations (Fig. 3) spread over the island, which are 'linked' and referenced to GPS stations on continental Greece. These were used to indicate possible tectonic deformation in Kefallinia, with respect to the 'stable' mainland. The GPS network was installed in October 2001 and their relative measurements were carried out in January and September 2003. The station locations (average spacing ~ 1 0 km) were selected according to the main geological structural units and to seismic activity concentration. The aim was to study the tectonic deformation triggered by faulting and by preand post-seismic activity. A station in the central part of Kefallinia (at Mt Ainos) was chosen as a reference (with fixed coordinates). In addition, the network was referenced to the Dionysos (DION) permanent GPS station in the Athens area, about 290 km east of Kefallinia, for monitoring larger-scale regional movements. Six geodetic receivers of WILD type (SR299 and SR399) were used for the GPS measurements. The Static Kinematic Software (SKI Pro, Version 3.2

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Fig. 1. (a) Sketch map of Europe-Africa plate interaction in the region of the Hellenic subduction. Active volcanic arc in the Aegean sea. Frontal thrusts indicated: the line with full triangles for the collision with Eurasia of Apulia, the submerged northern continental margin of the Ionian sea being part of Africa; the line with open triangles for the front of the accretionary complexes, in the west for the Calabrian one, in the east for the Hellenic one and the Mediterranean Ridge. In this subduction zone, the Hellenic Trenches, between full and dashed lines, are interpreted as outer arc basins made of upper plate material of the Aegean extensional region as well as a continental backstop to the accrefionary complex. Major active strike-slip faults are the North Anatolian fault, along which Anatolia is extruded, and the Cephalonia Transform Fault, the subject of the present study, which links collision with the western Hellenic subduction. After Sachpazi et al. (2000).

1999) of Leica allowed in situ processing and adjustm e n t of the GPS measurements. Post-processing of the GPS data was performed using the Bernese GPS Software Version 4.2 (Hugentobler et al. 2001), together with post-computed satellite orbits (downloaded from the International GPS Service, IGS). In this way, the error estimate of the coordinates of the stations on Kefallinla could be improved. Such adjusted values for the different periods of observations were considered, and an accuracy of 2 3 m m in the horizontal and 4 - 6 m m in the vertical c o m p o n e n t was finally achieved. The horizontal vector of deformation in the Kefallinia network, w h e n referenced to DION, has a NNE direction with an amplitude of ~ 2 0 m m ,

which is consistent with a clockwise rotation and with the regional tectonics of western Greece (Cocard et al. 1999). Concerning the DION station, an annual motion in a SSW direction of ,-~12 m m could be accurately defined. The final result is suggestive of a clockwise rotation of western Greece (Cocard et al. 1999) (Fig. 4). The local deformation was with reference to the Mt Ainos network. After the first remeasurement (January 2003), a small deformation of 4 - 1 0 m m was found in the horizontal component, and the vertical deformation range was 1 0 - 2 0 mm. The vector of the horizontal ground motion indicated a clockwise rotation around the main tectonic block of Mt Ainos, with small local discrepancies. The vertical

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Fig. 1. (b) The Kefallinla Transform Fault. The Kefallinla segment of the fault is marked as C and the Lefkada segment as L. The distribution of well-located seismicity (open circles) and epicentres of microearthquakes (filled circles), as well as the focal mechanisms of strong events are also shown. Figure and captions after Louvari et aL (1999).

movement show a general uplift of the island, with areas of subsidence to be associated with step-like faulting in the southwestem slopes of Mt Ainos and along the southern coast of the island. The DGPS results of the September 2003 remeasurement were strongly affected by the major 14 August 2003 earthquake (Fig. lc) that occurred on the island of Lefkada (Papadopoulos et al. 2003; Pavlides et al. 2004). Aftershocks appeared linearly concentrated along the northern part of Kefallinla. Stations located on this side of the island showed strong ground southeast motions, while the area was subsiding. Stations located further south appeared less affected, maintaining the same pattern as that inferred from the first remeasurement.

DInSAR techniques DInSAR has already proved its capability of providing images of ground surface deformation (Gabriel et al. 1989; Zebker et al. 1994; Massonnet & Feigl 1995, 1998; Peltzer & Rosen 1995; Hanssen 2001; Salvi et al. 2004; Wright et al. 2004). The technique

was used to plot interferograms, covering the investigated area, using radar images of the ESA satellites E R S - 2 and E N V I S A T . Differential interferometric images covering the period 28 September 1995 to 14 August 1998 seem to coincide with the ground deformation observed in the island, although the period is not exactly the same. The DInSAR image reveals deformations (Fig. 5) located in the northern part of Kefallinla, and mainly in the eastern sector, hence confirming the ground activity observed by DGPS measurements through time. A pair of SAR images was selected with a small baseline (Bp = 12 m) although with long time separation (ERS-1, 28 September 1995; and ERS-2, 14 August 1998). Ground deformation could thus be observed over a long period of time, during which no seismic event occurred. In general, the results of this analysis are consistent with the ground deformation recorded on the island by means of DGPS techniques. In more detail, every 360 ~ fringe circle represents one predefined great difference Az (called altitude of ambiguity) for every fringe of the interferogram. It can be

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Fig. 1. (c) The Lefkada earthquake, 14 August 2003, Mw = 6.2. Two clusters operated simultaneously. The first Mw = 4 event unambiguously located in the southern cluster occurred 9 min after the main shock. After Zahradm'k et al. (2005). calculated as a function of the wavelength of the radar signal, of the altitude of the satellite, of the angle of the signal, and of the perpendicular distance (Bpe~p) between the two orbits. In the present application, Az is ~ 4 9 5 m. Therefore, every fringe represents ~ 4 9 5 m in height.

Coherence characterizes the quality of the interferogram. In the present application, the coherence for the larger part of the island was quite low, leaving only selected areas with good coherence. The long temporal separation and the dense vegetation that covers most of the island are mainly

68

Fig. 2. (a) Geological map of Kefallinla.

M. POSCOLIERI E T AL.

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Fig. 2. (b) Map of structural blocks of Kefallin]a.

responsible for such poor coherence. In the Lixouri peninsula, in the west of the island, the vegetation is not dense. However, the poor coherence resulted from the geological features, characterized by loose formations and strong erosion.

The areas of good coherence are limited to the NE and SE of the island. We can define three areas: (1)

N o r t h e a s t e r n area. One fringe equals 28 m m of

deformation along the line of sight and the one

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Fig. 3. Location of the 23 GPS stations operating on the island.

(2)

(3)

fringe is formed along the topography of the area. The height difference in the area that covers the deformation fringe is about 500 m and is almost equal to AZ. This evidence, combined with the tectonic characteristics of the area and the control of the GPS results, denotes that the fringe represents the topography of the area rather than ground deformation. Agia Efimia area. In this area, a small, roughly circular, fringe can be defined. The spatial coverage of this fringe does not coincide with the topography and is located in an area where a NNW-SSE faulting zone occurs. It is not easy to explain this feature in detail, mainly because of its limited extension. However, the ground deformation associated with this fault zone could be related to localized phenomena and/or to vertical subsidence, which occurs in this area, according to GPS data. Digaleto area. In this area we observe the most interesting fringes. The coherence is good and almost two fringes of ground deformation are found, although the height difference in the area is only ~ 3 5 0 m . In the Digaleto area the outcropping geological units are made of Cretaceous carbonate

rocks of Ammonitico rosso formation, schists of Middle-Upper Jurassic times, Evaporites and Breccias of the Lower Triassic. The zone of deformation is also limited by two E - W and N E - S W oriented faults.

DEM techniques The geomorphic characteristics of the island can be quantitatively defined by a landform classification procedure, based on the analysis of the local morphological setting (Parcharidis et al. 2001; Cavalli et al. 2003; Parmegiani & Poscolieri 2003; Adediran et al. 2004). Geomorphometric data were gathered by processing a raster DEM (20 m per pixel ground resolution) produced by digitizing contour lines of a 1:50,000 scale map. The method is based on the application of multivariate statistics to an eight-layer stack, which describes the topographic gradients, measured along the eight azimuth orientations of the neighbourhood of every DEM pixel. This approach permits a quick estimate of the spatial distribution of different types of slope steepness, and a discrimination of areas characterized by similar local geomorphologic settings. Hence, it is possible to focus on changes in shape,

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Fig. 4. DGPS measurements. Comparison between first and second determinations (plots in the first column), second and third (second column), and first and third (third column), respectively. The first row deals with horizontal displacements, and the second row with vertical motions. The third row shows the regional large-scale displacement compared to the mainland (DION station).

orientation and steepness, and to stress the impact of erosional and tectonic processes on the overall relief. The classification technique chosen for processing gradient values was an unsupervised cluster analysis technique, ISODATA (Tou & Gonzales 1974; Hall & Khanna 1977). The following input parameters were chosen for the application of this multivariate procedure: 15 classes, threshold percent change of 1.0%, and 25 maximum iterations. The resulting classification map (Fig. 6) assigns every class a given grey shade. Classes were later statistically analysed by computing mean and standard deviation of the eight layers, which represent the elevation differences of every DEM pixel with respect to the neighbourhood. In addition, after calculating slope and aspect values from the same

DEM, the mean and variance of these parameters and of height were computed for all 15 ISODATA classes, and compared with the aforementioned eight-layer statistics, in order to give a correct geomorphic interpretation of the classification (Fig. 7). The preliminary analysis and interpretation of the geomorphometric image (Fig. 6), compared with the geological and tectonic block map (Fig. 2b), show a good correspondence with the geostructural setting of the area.

AE techniques and soil exhalation Since 1 February 2003, a multifrequency AE station has been in operation in Kefallinla, at a very central

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Fig. 5. Three-dimensional representation of the interferogram obtained by processing a couple of ERS-1 and -2 SAR images covering Kefallinia, with the main fault system of the island superimposed.

location of the island, and a nearby well is being monitored for Radon (Rn) content. Every AE datum is a 30 s integrated signal at its respective frequency. Periods of anomalous AE activity were observed. Consistent with previous evidence from the Italian peninsula (Paparo et al. 2006, this volume), they denote microdeformations preceding a seismic crisis. The HF AE (at 200 kHz) are correlated with Rn exhalation from the well. The rationale of the analysis is hereafter briefly explained. The AE signal is released whenever some atomic bonds yield within a crystal structure, and it propagates through the entire solid probe within which the crystals are embedded. Because of this, our AlE recorder should be located on top of a rocky outcrop, which is the terminal of a huge natural probe of unknown extension underground. Whenever a tectonic event makes the orientation of the natural probe change, the conspicuous crustal stress that is thus generated causes some AE release. The intensity of the AE recorded signal depends on the (unknown) efficiency of the waveguide that transfers the AE signal from its source through the detector. The HF AE will be the first

observed, corresponding to the yield of some lesser and tiny pores of the solid body. As soon as such pores coalesce into comparatively larger pores, some progressively lower frequency AE will be observed. These AE of progressively lower frequency will be later followed by mechanical vibrations of still lower frequency, until the seismic roar is heard, the vibration of mechanical structures occurs, and finally the seismic shock occurs. An earthquake, however, is not just a catastrophe that causes damages whenever some mechanical vibrations happen. It is, rather, a complex crustal phenomenon that must be investigated in all its aspects, beginning from its atomic level that releases some HF AE until the occurrence of the 'catastrophe'. In addition to the AE signal per se, which reveals the occurrence of some change in the statics and tectonics of the area, implying a time variation of the crustal stress, fractal analysis of the time series of the AE signals (see Appendix, page 59, for details) gives an effective indication of the state o f fatigue of the solid rocks that are releasing the AE. (Note, 'When the resultant of all the forces

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Fig. 6. Geomorphometric map of the Kefallin]a island. For explanation see Figure 7.

in a body is zero the body is said to be in equilibrium. This will be the case if it is at rest or moving with constant speed in a straight line. Both of these cases are grouped under the common heading of problems in statics' (Sears 1950, p. 15). Consider some huge rock, and it changes its orientation with respect to the local gravity vector. The internal stress distribution

shall change accordingly, eventually causing a release of AE.) In fact, the intensity of the AE release depends (although not linearly) on the intensity of the applied stress. In contrast, the response of a material is different depending on its ageing, which depends on its fatigue. Such property is revealed by the type of time sequence of the observed AE signals. However, it is impossible to

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Fig. 7. Height/slope/aspect mean values and morphostructural interpretation of the elevation differences between every pixel and the eight neighbours, for the 15 classes recognized in this study.

detect very feeble AE signals. It is reasonable to guess that the kind of temporal sequence associated with a given state of fatigue remains invariant starting from the undetectable and very feeble AE signals through their more intense and recordable AE impulses. Therefore, we have to transform the original AE record into a sequence of impulses that is, in the language of the mathematician, in order to recognize the fatigue of the medium, we have to consider a point-like process. (See Appendix, page 59, for details.) This was also shown by investigating several laboratory specimens of

different kinds (investigations not reported here). As far as the geophysical applications are concerned, a great advantage was obtained from the possibility of comparing AE records collected from different geological settings, dealing either with crustal stress originated by some external (tectonic) cause (such as typically occurs in the case of earthquakes), or by some endogenous source (such as occurs in geothermal or volcanic areas). Figure 8 shows the results for the Kefallin~a island. The AE records plotted are the smoothed weighted running average over _ 12 hours. (Note,

CRUSTAL STRESS AND SEISMICITY IN GREECE

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M. POSCOLIERI ET AL.

we carried out a running average over a total time lag of 24 hours, by applying a triangular filter, with maximum weigh at the centre of the time interval, and weigh linearly decreasing to zero from the entry time instant t, to t - 12 hours and to t + 12 hours, respectively.) Compared with previous case histories from the Italian Peninsula (see Paparo et al. 2006, this volume), a remarkable feature is the large scatter of the fractal dimension Dt v. time for both HF and LF AE (Fig. 8c, d). The LF AE series is more limited in time, because of the failure of the system as a result of a flood. The steep increase of HF AE in February 2003 seems to correlate with the large Rn maximum (Fig. 8a, e). The HF AE peak in the middle of June 2003 (Fig. 8a) ends when the peak of LF AE starts (Fig. 8b), and both peaks are followed by a Rn peak observed at the end of July 2003 (Fig. 8e). The HF AE (Fig. 8a) displays some activity. The fluctuations reveal that the AE sensor was unable to recover during the time lag elapsing between every two subsequent AE signals. That is, the sensor experienced a very rapid sequence of AE signals. This supports the fact that the entire region was presumably subject to some relevant stress propagation during the entire time interval. The HF AE spikes denote, presumably, some local features associated with the yield of microstructures at a site comparatively close to the AE recording site. Another possibility is that the AE source was at a comparatively great distance, but the AE source was much more intense. This second possibility could be suggestive of a correlation between AE and the Lefkada earthquake (Fig. lc), and the subsequent peaks could be better explained according to the first mechanism. In more detail, the system after the Lefkada earthquake remained perturbed during the entire period in which aftershocks were observed, and it shifted to an apparently quiet trend when the aftershocks ceased. Then, according to the HF AE, the system remained in a comparatively quiet state for a while, until January 2004, when it started to be perturbed anew by some activity shown by a scattered and increasing trend. The LF AE plot in Figure 8b shows some activity during June-July 2003, preceding the Lefkada earthquake by approximately two months. According to the same rationale, some lesser activity in January to March 2004 could be associated with four events of Mw ~ 4 with an epicentre in northern Kefallin~a. Shortly after this occurrence, an increasing trend occurred after 18 April 2004, showing that the region had very frequent AE signals, and the sensor could not recover between any two subsequent signals. Independent of their specific physical interpretation (which, in any case, appears still fairly premature), three different observations can be made from the LF AE plot in Figure 8b. First,

there is essentially no trend except during the very last period of records, when it increases. Second, a simple scatter of points is superposed, almost like noise (however, is not noise) and this should have a physical implication. Third, the scatter or 'noise' seems to increase significantly, being in some way approximately correlated with the amplitude of the increasing trend. During May 2004 a remarkable amount of seismic activity was concentrated in the northern region of Z~kinthos. The large LF AE peak observed shortly after 9 June 2004 (Fig. 8b) could be an object of serious concern. This much larger peak could denote that either the AE source is much closer to the AE sensor than in 2003, or that a new earthquake is forthcoming, at a comparatively great distance, but of some stronger intensity. Obviously, this is not a prediction. One could tentatively consider this feature as being one possible precursor, suitably correlated with other evidence by means of other techniques. On the other hand, such a phenomenon could be interpreted differently. The HF AE trend (Fig. 8a) can be considered as separating the long-range from the short-range trend. The short-range HF AE trend displays some apparently erratic variations, which, as mentioned above, ought to reflect local stress and microdeformation, although displaying the aforementioned correlation with aftershocks. The long-range HF AE trend clearly displays a possible yearly variation, which is not significant p e r se, because, strictly speaking, several years of records ought to be available. However, the same yearly trend, although eventually phase shifted, is observed also in the Italian Peninsula. If this inference is confirmed, an interpretation has to be attempted. Considering the conspicuous difference in tectonic settings between Italy and Kefallinla, a tentative possible guess is that a yearly wave of crustal stress is steadily involving the entire central and western Mediterranean area, potentially associated with astronomical forcing. For the time being this is mere speculation, and a correct interpretation can be given only by considering HF AE records collected at several different locations, perhaps operated non-simultaneously, in order to map the apparent yearly trend, and its amplitude and relative phase. With this perspective of the HF AE, the LF AE appears comparatively less interesting, as no apparent or suspected yearly trend is observed. In general, however, the AE records seem to provide a significant and unprecedented monitoring of the evolution of the crust during its preparatory stage before the eventual occurrence of an earthquake. It appears to be a useful technique also for diagnosing the temporal evolution of crustal stresses, independently of whether an earthquake is going to occur or not.

CRUSTAL STRESS AND SEISMICITY IN GREECE

Conclusions With respect to the AE technique, Kefallinla and the Italian peninsula, although having very different tectonic settings, display some interesting analogies, although every realistic inference on crustal stress evolution requires arrays of simultaneously operated AE recording stations. More in general, this would allow us to better explore the potential of fractal analysis of AE time series signals in order to furnish important information on the state of fatigue of stressed crustal rock volumes. Regarding the other techniques that are considered in the present study, their results appear to be in agreement with each other, although, as expected, no final conclusion seems as yet possible. The 'largescale' information derived from satellite remote sensing or from DEM analysis lacks any adequate time resolution. In contrast, the field records (AE and soil exhalation) have a good time resolution, although they are point-like, and one needs an array of recording points to obtain information valuable for geophysical inference. Our results might show, however, that the crustal stress in Kefallin]a island propagates and interacts with (or, perhaps, it contributes to generating) the highly complicated faulting network responsible for the seismicity of the area. The degrees of freedom of the system are, however, too large, and no simple or intuitive model can yet be proposed from the presently available database.

References ADEDIRAN,O. A., PARCHARIDIS,I., POSCOLIERI,M. & PAVLOPOULOS, K. 2004. Computer-assisted discrimination of morphological units on northcentral Crete (Greece), by applying multivariate statistics to local relief gradients. Geomorphology, 58, 357-370. CAVALLI,R. M., FUSILLI, L., PASCUCCI,S., PIGNATTI, S. & POSCOLIERI, M. 2003. Relationships between morphological units and vegetation categories of Soratte Mount (Italy) as inferred by processing elevation and MIVIS hyperspectral data. In: BENES, T. (ed.) Proceedings of 22nd EARSeL Symposium and General Assembly, Prague (Cleck Rep.), 4 - 6 June 2002, Millpress, Rotterdam (Netherlands), 573-579. COCARD, M., KABLE, H. G. ET AL. 1999. New constraints on the rapid crustal motion of the Aegean region: recent results inferred from GPS measurements (1993-1998) across the West Hellenic Arc, Greece. Earth and Planetary Science Letters, 172, 39-47. GABRIEL, A. K., GOLDSTEINR. M. & ZEBKER, H. A. 1989. Mapping small elevation changes over large areas: Differential radar interferometry. Journal of Geophysical Research, 94(B7), 9183-9191.

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HALL, D. J. & KHANNA, D. 1977. The ISODATA method of computation for relative perception of similarities and differences in complex and real data In: ENSLEIN, K., RALSTON,A. 8~;WILF, H. S. (eds) Statistical Methods for Digital Computers 3, John Wiley Pub., New York, 340-373. HANSSEN, R. F. 2001. Radar Interferometry. Data Interpretation and Error Analysis. Kluwer Academic Publishers, Dordrecht. HUGENTOBLER, U., SCHAER, S. & FR1DEZ, P. 2001. Bernese GPS Software Version 4.2 Documentation. Astronomical Institute University of Bern, Switzerland. LOUVARI, E., KIRATZI, A. A. & PAPAZACHOS,B. C. 1999. The Cephalonia transform fault and its extension to western Lefkada island (Greece). Tectonophysics, 308, 223-236. MASSONNET, D. & FEIGL, K. L. 1995. Discrimination of geophysical phenomena in satellite radar interferograms. Geophysical Research Letters, 22, 1537-1540. MASSONNET, D. & FEIGL, K. L. 1998. Radar interferometry and its application to changes in the Earth's surface. Review of Geophysics, 36, 441-500. PAPADOPOULOS,G. A., KARASTATHIS,V., GANAS,A., PAVLIDES,S., FOKAEFS,A. 8~;ORFANOGIAUNAKI,K. 2003. The Lefkada, Ionian Sea (Greece), shock (Mw 6.2) of 14 August 2003: Evidence for the characteristic earthquake from seismicity and ground failures. Earth Physics Letters, 55, 713-718. PAPARO, G., GREGORI, G. P., POSCOLIERI, M., MARSON, I., ANGELUCCI, F. & GLORIOSO, G. 2006. Crustal stress crises and seismic activity in the Italian peninsula investigated by fractal analysis of acoustic emissions, soil exhalation and seismic data emission (AE), soil exhalation and seismic data. In: CELLO, G. & MALAMUD,B. D. (eds) Fractal Analysis for Natural Hazards. Geological Society, London, Special Publications, 261, 47-61. PARCHARIDIS, I., PAVLOPOULOS, A. & POSCOLIERI, M. 2001. Geomorphometric analysis of the Vulcano and Nysiros island: clues of the defintions of their volcanic landforms. In: GIOVANNELLI, F. (ed.) Proceedings of the International Workshop "The Bridge between Big Bang and Biology", Stromboli (Messina, Italy), 13-17 September 1999. CNR, President Bureau, Special Volume, 310-320. PARMEGIANI, N. & POSCOLIERI, M. 2003. Studio dell'impatto antropico sull'assetto morfologico di un'area archeologica. In: Proceedings of the 7th ASITA National Conference "L'informazione territoriale e la dimensione tempo", Verona (Italy), 28-31 October 2003, 2, 1569-1574. PAVLIDES, S. B., PAPADOPOULOS,G. A., GANAS, A., PAPATHANASSIOU, G., KARASTATHIS, V., KERAMYDAS, D. & FOKAEFS, A. 2004. The 14 August 2003 Lefkada (Ionian Sea) earthquake. In: Proceedings of the 5th International Symposium on Eastern Mediterranean Geology, Thessaloniki, Greece, 14-20 April 2004, Paper T5-34, 1-4.

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PELTZER, G. & ROSEN,P. 1995. Surface displacements of the 17 May 1993 Eureka Valley, California, earthquake observed by SAR interferometry. Science, 268, 1333-1336. SACHPAZI,M., HIRN, A. ETAL. 2000. Western Hellenic subduction and Cephalonia Transform: local earthquakes and plate transport and strain. Tectonophysics, 319, 301-319. SALVI, S., GANAS, A. ET AL. 2004. Monitoring longterm ground deformation by SAR Interferometry: examples from the Abruzzi, central Italy, and Thessaly, Greece. In: Proceedings of 5th International Symposium on Eastern Mediterranean Geology, Thessaloniki, Greece, 14-20 April 2004, Paper T7-17, 1-4. SEARS, F. W. 1950. Mechanics, Heat and Sound. Addison-Wesley, Reading, MA. Tou, J. T. & GONZALES,R. C. 1974. Pattern Recognition Principles. Addison-Wesley, Reading, MA.

UNDERHILL,J. R. 1989. Late Cenozoic deformation of the Hellenic foreland, Western Greece. Geological Society of America Bulletin, 101, 613-634. WRIGHT, J., PARSONS, B., ENGLAND, P. C. FIELDING, E. J. 2004. InSAR observations of low slip rates on the major faults of western Tibet. Science, 305, 236-239. ZAHRADNfK, J., SERPETSIDAKI, A., SOKOS, E. & TSELENTIS, G.-A. 2005. Iterative deconvolution of regional waveforms and a double-event interpretation of the 2003 Lefkada earthquake, Greece. Bulletin of the Seismological Socie~ of America, 95, 159-172. ZEBKER, H. A., ROSEN, P. A., GOLDSTEIN, R. M., GABRIEL, A. K. & WERNER, C. L. 1994. On the derivation of coseismic displacement-fields using differential radar interferometry - the Landers earthquake. Journal of Geophysical Research, 99(B10), 19617-19634.

Nonlinear Science issues in the dynamics of unstable rock slopes: new tools for rock fall risk assessment and early warnings JIl~I Z V E L E B I L a, M I L A N P A L U S 2 & D A G M A R N O V O T N A 3

aGeo-tools NGO, U Mlejnku 128, 250, 66 Zdiby & Czech Geological Survey, Kldrov 3, 118 21 Prague 1, Czech Republic (e-mail: [email protected]) 2Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod voddrenskou v ~ f 2, 182 07 Prague 8, Czech Republic 3Institute of Atmospheric Physics, Academy of Sciences of the Czech Republic, Bo(n{ II/1401, 141 31 Prague 4, Czech Republic Abstract: Time series of displacement data from unstable rock slopes contain 'hidden' information about the dynamics of slope failure. This information cannot be found when using the current linearly causal paradigm based on analytical methods, but is revealed when numerical and graphical methods from the toolbox of the Nonlinear Sciences are applied. The occurrence of fractal patterns, which suggests a qualitative difference between intrinsic slope movement dynamics of time series from the near-to-equilibrium and the far-from-equilibrium dynamical states of slope failure systems, is an example of such a 'hidden', diagnostically important indicator. It helps to identify the stage of immediate danger of rock fall occurrence, just in time to launch an efficientearly warning. Phase portrait and correlograms of time series proved to be suitable for earlier revelation of transitions from the near-to-equilibrium to the far-from-equilibrium dynamical states, as well as for helping to distinguish between intrinsic slope movement dynamics and climatically driven reversible deformation activity.

The holistic paradigm of the nonlinear complex character of natural systems is gaining credit in various fields of geoscience (e.g. Turcotte 1997, 2000; Phillips 1999; Viles 2001; Sivakumar 2004). Natural geosystems are in fact very complex and highly interactive; their parts interplay with each other, forming 'a network of networks', with the possibility of surprising new qualities emerging in their behaviour or attributes. These qualities could not be deduced simply from the quality of the interacting parts, because the whole geosystem is more than just a sum of its parts. In addition, findings about such systems are contextually dependent. They can be fixed in being causal or random at the same time, according to the specific relationships that are studied within them, or because of the relationship chosen according to the spatial-temporal scale used. The problems that arise when we try to understand these very complex systems using linearly analytical tools of 'classical' physics are well known, as discussed in nearly every book dealing with nonlinear dynamics and dynamical systems (e.g. Cohen & Stewart 1995; Bar-Yam 1997; Kantz & Schreiber 1997; Meakin 1998). Here, we would like to stress that current 'classical' methods fail not only in their 'holy aim' to be able, if initial conditions within the system are known, to

predict future behaviour precisely, but sometimes also in an adequately realistic description of the actual state of the given geosystem. This situation was recognized in the early stages of research in rock slope failure (cf. Terzaghi 1962; Mtiller 1980). Since then, two different approaches have been developing side by side in the field of engineering geology. The first one, known as a 'classical' geomechanically based approach, aims at elaborating complex models that take into account more and more factors and processes (e.g. Poisel & Preh 2004; Poisel & Roth 2004). The other approach is instead based on a holistic model of dynamics of unstable slope (i.e. on the description of the behaviour of that slope). The quest for such a model was started by Bjerrum & Jorstadt (1968) in their famous paper about rock falls and their forecasting in Norway. They called their approach the 'Observational Method', because it was based on assessment of slope monitoring results. To fix an actual degree of rock slope instability, the authors recommended a scaled list of symptoms, characteristic dynamical patterns of displacement and deformation phenomena, whose scaling would correspond with the different stages of preparation of catastrophic slope collapse. Since then, such 'empirical-phenomenological models' of temporal development of slope

From: CELLO,G. & M~LAMUD,B. D. (eds) 2006. FracmlAnalysisfor Natural Hazards. Geological Society, London, Special Publications, 261, 79-93. 0305-8719106/$15.00 9 The Geological Society of London 2006.

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movement activity have been used to assess instant slope instability and temporal prognostication of rock fall occurrence (Saito 1969; Voight & Kennedy 1979; Fuzukono 1984; Zvelebil 1985, 2004; Rochet 1992; Zvelebil & Moser 2001). Those models were applied mainly to interpret the geometric features of data curves in common, time-deformation, Cartesian plots of monitoring time series. Limitations of those empiricalphenomenological models have, however, been questioned since they first came into use (e.g. Zvelebil 1996; Moser et al. 2002; Moser 2003). Nonlinear analysis and modelling of time series data offer an opportunity to overcome those limitations. More than 16 years ago, Zvelebil (1984) started to compare the complex hierarchical patterns of time series from dilatometric (thermal expansion/dilation) monitoring of rock cracks, according to the concepts and methods of selforganizing systems such as those discussed in Nicholis & Prigogine (1977) and Prigogine & Stengers (1984). He arrived at the conclusion that 'Part of the important information which is embedded in monitoring time series, is hidden. When current linear-based methods are employed it mimics itself as a seemingly random noise (e.g. Rfi~ek & Zvelebil 1993). Hence, we should look for new more appropriate methods, and the toolbox of nonlinear dynamics seems to be a reasonable choice' (Zvelebil 1996). The present paper deals with the preliminary results of a joint challenge for an engineering geologist, a mathematician, and a physicist to find new, mathematically rigorous tools for better handling of monitoring data from unstable rock slopes.

Search for hidden information In this paper, the term 'nonlinearity s e n s u stricto (s.s.)' is defined, in a mathematical sense, as dynamics that cannot be reduced to a standard linear autoregressive model or its static, possibly nonlinear transformation. A special example of such processes can be deterministic chaos. 'Nonlinearity s e n s u lato (s.1.)' implies a wider, hence vague meaning; it has been introduced in the field of Nonlinear Science to provide a summarizing label for the very specific behaviour features of complex systems that are difficult to elucidate within the ordinary frame of linear paradigms (i.e. nonlinearity s.s., emergency, self-affinity, self-organization, selforganized criticality, etc.) (e.g. Bar-Yam 1997). In our search for hidden information, we used data from a regional monitoring network of sandstone rock slopes in northwestern Bohemia (e.g. Zvelebil 1989, 1995; Zvelebil & Park 2001). The network started operating in 1979 in order to

monitor the Czech-German traffic corridor through the deep canyon of the River Labe. It has been gradually expanded to encompass slopes above settlements and tourist paths in areas of the highest rock fall risks within the National Park Bohemian Switzerland. From the wide spectrum of methods available for measuring rock slope deformations, dilatometry was chosen. Using that method, systematic measurements of changes in length (displacements) of measuring lines placed across rock cracks, have been carried out. Nowadays, the network spreads over 327 rock volumes with more than 900 sites where dilatometric measurements of relative displacements along rock cracks are currently measured (cf. Vafilovzi & Zvelebi12005). The longest monitoring time series span over 25 years. The data set includes nearly all the developmental stages of sandstone rock slope instability. The quality of time series differs according to the monitoring techniques; these include manual measurements, carried out with a portable rod dilatometer, which cover the longest time interval and the broadest spectrum of developmental stages occurring in the course of a rock fall preparation. Unfortunately, the quality of the data suffers from irregular sampling and from variations in the sampling time interval, which ranges from a few days to one month. This irregular sampling forced us to modify well-established methods of Nonlinear Dynamics (e.g. Kantz & Schreiber 1997), or to re-sample the data in order to perform the analysis. The series used for the analysis included some 480 to 612 samples and the available time span was from January 1984 to June 2001. Besides the time series obtained from manual dilatometry, we also analysed the results supplied by automatic acquisition systems. They include from 13,000 to 123,000 samples taken at regular frequencies of 5 or 10 minutes. Those time series spanned from 3 to 14 months. Slope monitoring signals consist, as do all signals from natural dynamical systems (e.g. Perry et al. 2000), of a mixture of coexisting and interacting dynamics. For this reason, signals relating to rock mass failure have to be distinguished from displacements and deformations of different origin. There is quite a long list of displacements/deformations due to causes other than slope movements resulting from rock mass failure (cf. Zvelebil 1989); it includes mainly data due to reversible responses of the rock mass to perturbations by the external environment. The most important one is represented by changes of rock-block volumes due to temperature variations. The patterns of these thermal dilations of rock blocks correspond to the hierarchically structured system of climatic cycles, from the diurnal and seasonal up to ones taking many years (Zvelebil 1995). The whole polygenetic assemblage

NONLINEAR SCIENCE ISSUES IN ROCK SLOPES of those reversible responses to perturbations by the external environment will be called the 'standard activity' (SA) in the following sections. In order to minimize errors in distinguishing between signals due to SA manifestations and those peculiar to rock slope displacement, all data without any detectable evidence of slope movement activity were omitted from the analysis, and the remaining time series were divided into two groups: one representing the 'near-to-equilibrium' signals and the other representing the 'far-from-equilibrium' states of unstable slope systems. The 'near-to-equilibrium' (NTE) series were recorded on slopes that exhibit irreversible long-lasting slope movements, but where no patterns indicating rapid slope collapse were identified (e.g. Figs l a, 5a-c). The 'far-fromequilibrium' (FFE) series were obtained from recently collapsed slopes only (Figs 2a, 3a, 4a, 5d, e).

Graphical tools: phase space portrait and correlogram A major achievement of Dynamical Systems Theory has been that of bringing us back to geometry as an important and rigorous tool for studying system dynamics (Abraham & Shaw 1992). The geometrical patterns of common displacementtime plots have played a prominent role in data interpretation using current empirical-phenomenological models (Voight & Kennedy 1979; Fukuzono 1985; Zvelebil 1985, 1996; Zvelebil & Moser 2001). These types of plots of rock slope displacements are represented by Figures la, 2a, 3a, 4a, and 5a. In this paper, two other ways to analyse time series, phase portraits and correlograms, were tested. Although these are quite common tools in Dynamical Systems and Harmonical analysis, they have not yet been used in the field of slope monitoring. Phase portraits of 'raw', that is, non-filtered monitoring time series embedded in two- and three-dimensional phase space, have been found to be quite appropriate in fulfilling the crucial task of detecting the transition from NTE to FFE dynamics. A phase space is a vector space, in which any point specifies the instant state of the given system and vice versa. It is a powerful tool for giving a geometrically synoptic display of characteristic patterns of very complex behaviour that, as for non-deterministic systems, can be displayed by a huge (possible infinitive) set of states and some kinds of transition rules that specify how the system may proceed from one state to the other (Kantz & Schreiber 1997, pp. 30-31). As the behaviour of the system develops in time, a sequence of its state points clusters into a geometrical entity of state trajectory within phase space.

81

Any geometrically regular pattern that emerges by clustering of system state trajectories in phase space corresponds to certain regularities in behaviour of the given system. Geometrical patterns of the NTE state are shown in Figure 1 and the FFE ones in Figure 2. The phase portraits are more synoptic than the current displacement- time plots of the series in question, especially when very long time series are being studied. In the phase portrait, all regular patterns which should otherwise be laboriously traced along the whole length of such time series, are 'compressed' in one section of phase space, making an attractor image, that is, a distinct geometrical pattern that represents the whole group of similar but not necessarily equal types of behaviour of the given system. Any transition from the given set of patterns of behaviour to some other type of behaviour corresponds to the system state trajectory that is, usually quite distinctly, heading out of the basin of a given attractor (compare Fig. lb with Fig. 2b). In Figure 2b, the heading-out trajectory was interpreted as an indication of a phase shift from the current NTE state (marked ~1, near the origin in Fig. 2b) to an FFE type of system behaviour (marked a). It was possible to detect this N T E FFE shift even 13 months before (Fig. 2a). The inverse development is marked 82 in Figure 2b. In Figures 3a and b, there is another representation of the FFE to NTE shift, which occurred in the course of new local crushing at the toe of a high rock wall. After an initial, high activity of new joint spreading, a stress rearrangement towards an inner lessdisturbed zone occurred, resulting in a gradual low-down of the displacement along the new joint. Correlograms of the time series can help us to identify where displacement patterns from different parts of the slope display interrelated features, that is, which series are produced or influenced by the same process. In this paper, the variant called XY plot in MATLAB usage is adopted. Data from different time series or of the same type (but from different places) or of different types (e.g. displacement and temperature) from the same place, are plotted in two 2D or 3D plots. Any relationship between those time series is characterized by a specific pattern, which may be quantified by measuring coupling and synchronization (Palu~ et al. 2001; Palu~ & Stefanovska 2003; Pikovsky et al. 2003). In Figures 4a-b, we show data from different parts of a large, unstable rock pillar. The high degree of synchronization of movement events can be spotted between records from the uppermost scarp and the lower frontal and toe parts of the pillar (Fig. 4b). The results of the analyses shown in Figure 4 support our preliminary assumption about the existence of deep-seated phenomena affecting the whole rock mass.

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Fig. 1. (a) Regular and (b) phase space portrait plots of the same time series representing the medium stage of preparation of a rock fall, that is, the 'near-to-stability' (NTS) state of the system. The data refer to a twenty-year record of a slowly sinking and toppling rock block with a volume of 1600 m 3, which forms the toe of a 100 m high rock wall. It may be observed that it is sometimes difficult to assess the intrinsic dynamics of relative displacements between rock blocks by slope stability failure using time-displacement data, as the patterns of Figure la are distorted by some underlying 'noise' (here named 'standard activity', SA). SA may be a result of (a) seasonal activity with amplitude of about 3 mm, generated by volume changes of rock blocks as a result of temperature variations, and to (/3) an almost cyclic activity, with a duration of 10-11 years; this type of SA may also be of climatic origin. The intrinsic dynamics of slope stability failure (~) show instead a long-lasting linear trend with a gradient of 0.1 mm/year. Other types of displacements of dubious origin (6) include a 3-5 year cycle, which may be simulated by a simple vector addition of a 10-11 year almost cyclic SA with linearly increasing irreversible displacement, or may be at least partially caused by changing the rate of irreversible slope movements. A two-dimensional phase space portrait of the same regularly resampled data set is shown in (b). It may be seen that the resulting pattern mimics a hypothetical attractor. Six loops of state trajectories shifted by translation gliding (a) along the symmetry axis of the attractor correspond to the periods of increased activity of irreversible movements, and all denser areas (/3) correspond to periods of relative calm.

Numerical tools: distributions and temporal correlations Full details of mathematical scrutinizing of given monitoring time series are presented in Palu~ et al. (2004). Here, we summarize our main results. For this type of analysis, the basic division into two groups (the N T E and FFE groups) was retained; however, the majority of time series had to be excluded from the analysis because they were incomplete (gaps too large in the records). Only four FFE and five NTE series were suitable for analysis (see Fig. 5 and Figs la and 2a). The chosen time series

were also regularly resampled upto 1024 samples. Because the raw dynamics of the series was clearly dominated by atmospheric influences, mainly by temperature (visible demonstrations of SA), the atmospheric variables were also considered in our analyses. With this aim, meteorological data were resampled. Our numerical tools include techniques of uniand multivariate surrogate data with simple phase randomization, fast Fourier transform and the Schmitz & Schreiber (1999) construction method, and the method of information-theoretic functionals-redundancies of Palu~ (1995a, b). The

NONLINEAR SCIENCE ISSUES IN ROCK SLOPES

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0.2 o

0 -0.2

..

.

,o.v,,.oo

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-0.2

0

0.2

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Fig. 2. (a) Time-displacement plot and (b) phase portrait of time series from site L2 show the transition from (a) near-to-stability (NTS) to (/3) far-from-stability (FFE) states of a slope system. The time-displacement plot (a) enabled us to empirically detect markers of the NTE-FFE transition from November 1997, whereas the 2D phase portrait diagram (b) allowed us to detect this same information one year earlier, when a state trajectory heading out of the NTE attractor was clearly defined.

(a)

~

~.~

12.0 E

E

8.0

4.0

~

//

"

1III (~

~1/t 2000

4000

6000

T (days)

(b)

X (DT)

Fig. 3. (a) Time-displacement plot and (b) 3D phase space portrait of the FFE (a) to the NTE (fl) transition in the rock slab of Figure 1. From the phase portrait diagram, one may see that there is a general shift of the system state trajectory towards the NTE attractor, before it finally sets inside the/3 space.

84

J. ZVELEBIL E T AL.

Fig. 4. (a) Displacement patterns of time series from three different positions ($5, $6, $7) within a rock pillar with a volume of 3000 m 3. (b) Correlograms of the same data set. The high degree of synchronization (a) between data from the uppermost section of the pillar ($7) and those from the lowermost section ($5) suggests that slope movements are induced by deep-reaching processes. The lack of synchronization (13) between data from section ($7) and ($6) was instead interpreted as resulting from an independent process occurring near the surface and affecting smaller rock volumes. Minor synchronization events (marked by the arrows in/3) have been related to the rock mass response to climatic perturbations.

NONLINEAR SCIENCE ISSUES IN ROCK SLOPES

E

E ,,--,,., o

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2 1 0 -1 -2 1

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0 -1

(b) 3 E

& o

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1 0 -1

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(e) 1985

, 1990 TIME [YEARS]

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2000

Fig. 5. Dilatometric measurements of relative displacements observed across cracks in sandstones: NTE dynamics (a-c), FFE dynamics (d, e). intrinsic slope dynamics of both the NTE and FFE time series were characterized using analyses of the residuals (Fig. 6) obtained from the dilatometric series after removing meteorological influences. Those residuals were obtained by triple linear regressions (DATAPLORE SW Package, 2006). The plots of empirical probability for amplitudes larger than the given value were constructed for the distribution tests. In order to study the dynamics and temporal correlations, the power spectra of residuals were calculated (DATAPLORE SW Package, 2006). Scaling of the distribution of fluctuations and of the distribution of energy over the power spectrum, as well as a possible scaling of fluctuations in their temporal evolution were studied using a detrended fluctuation analysis (Peng et al. 1994, 1995; Goldberger et al. 2000). Using the listed tools, the following results were obtained. (1)

Nonlinear dynamics s.s. The necessary conditions for proving the presence of nonlinear

(2)

dynamics s.s. were not fulfilled (Figs 7-10). Our finding predominantly concerned the strong influence of atmospheric variability and seasonality on monitoring the time series, as these were mainly expressed by their SA component. The influence proved to be linear, but, at the same time, not trivial. Note that two previously unknown thne lags of 100 and 123 days were found from regressions of the annual cycle of atmospheric temperature dynamical features onto the dilatometric series. Nonlinearity s.l. This was detected in the intrinsic slope dynamics of the FFE series, but not for the NTE ones. There is a qualitative difference of correlation decay in the dynamics of the NTE and FFE series. The residuals from the NTE series possess nontrivial, but nevertheless linear dynamical features. They are non-Gaussian, asymmetrically distributed, fat-tailed (e.g. Malamud 2004; Malamud & Turcotte 2000) fluctuations with short-range correlations (Fig. 1 la).

J. ZVELEBIL ET AL.

86 T

"T

T

1 E E tm

0 -1

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2 1

0 -1

2O ~' o ~-

10 0

-10

80 6o

3O E E 20 n," 10

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1990 TIME [YEARS]

1995

2000

Fig. 6. Linearly detrended NTE (a) and FFE (b) time series of dilatometric measurements D, and time series relative to (c) atmospheric temperature T, (d) humidity H, and (e) precipitation R.

Nonlinearity s.1. could be considered for the FFE residuals. They are characterized by an asymptotic power-law distribution on the 'fatter' side of their non-Gaussian distribution (Figs l lb and 12). Its decay coefficients range between 4 and 5, that is, outside the range of stable Lrvy distribution 0 < / . ~ < 2 (Schertzer & Lovejoy 1991). For this type of fluctuation, the dynamics is intermittent and high-order moments diverge. Furthermore, the dynamics of FFE residuals possesses persistent long-range correlation of selfaffine processes (an occurrence of l/f, that is, of pink noise, see Turcotte 1989; Barrow 1995; Malamud & Turcotte 1999). Moreover, two scaling regions were consistently identified by both the spectral and the detrended fluctuation analyses. In time-scales between 4 and 11 weeks, the persistence is characterized by the spectral decay coefficient /3 ~ 2, which corresponds to a Brownian motion. Time-scales from 11 weeks to almost 2 years are described by the

spectral decay coefficient /3 ~ 1.5, which corresponds to a fractional Brownian motion. The information obtained by using this nonlinear approach for the study of unstable rock masses also poses new questions; some of them are discussed in the following sections.

Discussion Specific results There is a disproportion of spatial-temporal scales between monitoring records and the development of slope failure. The relatively short NTE time series represent merely point-like samples of the precritical stages of slope failure systems, whereas the FFE time series roughly match the critical precollapse stage. Unfortunately, in our case study, sampling of FFE time series was too coarse to reveal the finer details of their dynamics. Hence, our information is relevant only for dynamical

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20

40

60

80

100

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LAG [sample]

Fig. 7. Nonlinearity test of a detrended unstable dilatometric time series, using (b, d) mutual information I (X(t);

X(t + ~-)), and the check of the surrogate data using (a, c) linear mutual information L (X(t); Y (t + ~-)). The values of mutual information (a, b) from tested data (solid line), mean (dashed-and-dotted line) and mean -t- s.d. (dashed lines) of a set of 30 measurements are shown, as are the statistical differences in the number of standard deviations (s.d.) of the surrogates (c, d). 9

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Fig. 8. Testing nonlinearity in the relationship between atmospheric temperature and the detrended unstable dilatometric time series using (b, d) mutual information I (X(t); Y (t -t- ~')), and the check of the surrogate data using (a, e) linear mutual information L (X(t); Y (t + r)). See caption of Figure 7 for key.

88

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Fig. 9. Testing nonlinearity in the relationship between atmospheric temperature and the residuals of the multilinear regression of the detrended FFE dilatometric time series on the meteorological variables using (b, d) mutual information I (X(t); Y (t + ~-)), and the check of the surrogate data, using (a, c) linear mutual information L (X(t); Y (t + ~')). See caption of Figure 7 for key.

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100

40

60

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80

100

0

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100

Fig. 10. Testing nonlinearity in residuals of the triple linear regression of the detrended FFE dilatometric time series on the meteorological variables, using (b, d) mutual information I (X(t); Y (t + ~-)),and the check of the surrogate data using (a, c) linear mutual information L (X(t); Y(t + r)). See caption of Figure 7 for key.

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-3

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0

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90

J. ZVELEBIL ETAL.

patterns that are sufficiently robust to be fixed by a 14-day frequency sampling condition. On the other hand, these patterns reoccur so frequently that they may be spotted, at sufficient significance levels, within a time interval of about 15 years. Our results also show that the response patterns of slope failure systems, which represent a family of complex highly interactive catastrophic events, met our theoretically based expectations, as they allowed us to unravel previously hidden information, such as a time lag response of 100 and 123 days. On the other hand, the linear (even though nontrivial) nature of dilatometric time series modulation by climatic influence is rather surprising. In fact, it does not conform to the findings of other authors (e.g. Tausch et al. 1993; Kupfer & Cairns 1996; Phillips 1999), as climatic driving, or driving containing any climatic and therefore inherently chaotic component, implies the high possibility of unstable, chaotic elements in the dynamics of the system. In any case, the relatively simple linear form of climatic modulation does not a priori imply any theoretical restriction to improving the ability and reliability of filtering out the SA component from the monitoring signal. The question of whether the intrinsic NTE dynamics really possesses climatic driving or is a consequence of insufficient filtering of the SA component has still remained unanswered. Comparison of the dynamics of NTE time series with records from stable slopes where only the SA component is present should help answer this question. The reliability of fixed patterns to correspond with intrinsic slope failure dynamics is relatively greater for FFE series. In this case, dynamics includes fluctuations with hyperbolic intermittency and scaling spectra and is supposed to occur in response to the action of cascading, energy-transferring processes (e.g. Schertzer & Lovejoy 1991). The robust fitting of the distribution of FFE residuals can indicate the occurrence of a self-organizing process (Bak & Chert 1991; Jensen 1998; Turcotte 2000; Turcotte & Rundle 2002; Sornette et al. 2004). The existence of two scaling regions implies that the intrinsic fractal dynamics of FFE is scale-dependent. Therefore, the next step should be a fractal analysis on short time-scales using high-frequency time series from automated data acquisition systems. In any case, the qualitative difference between the NTE and FFE dynamics, as well as the geometrically distinct transition form NTE to FFE states and the occurrence of fractal patterns of time series residuals after SA filtering, seem to be quite important for further enhancement of the early warning issue. With this aim, these were successfully used for the safety evaluation of monitoring data during emergency remedial works in I-I~ensko village in 2002 (Zavoral 2002; VaY-ilov~i& Zvelebil

2005). The most recent case history of early distinguishing of rock fall danger, and a successful time-prediction of that rock fall occurrence in Kamenice River Gorge (Vafilov~i & Zvelebil 2005), has shown that correlograms of displacement time series from different monitoring sites, as well as those relating deformation and temperature changes from the same site, may be useful tools for monitoring data assessment. For this reason they are currently being introduced as a regular part of an integrated monitoring system for the whole territory of the Czech-Switzerland National Park. The method of displacement/displacement correlograms was also successfully applied to clear kinematics of slope movements endangering the Spi~ Castle, Slovakia (Ba~kova 2004; Vlfiko 2004). General implications

Sticking to current evaluation tools for monitoring data, which are based on linear reductionist paradigms, may result in biased or incorrect handling of slope dynamics analysis. The list of possible errors and misleading conclusions includes the following: (1) Overestimation of the proportion of random noise within the signal, accompanied by an inability to see the 'hidden' order. This is our topical case of hidden information masked by white noise. (2) The linear presumption; it is only the external influence that matters in changes of system dynamics. This disregards the possibility of dynamical changes due to the action of inner mechanisms of slope failure, as well as the existence of various responses, differing in their timing, of the slope system to the same perturbations. This is a cardinal phenomenon to be considered in every triggering-factor study. (3) The complex nature of slope failure and the variety of local conditions dictate that fully quantitative specifications are practically impossible, and that even location specifications can be exceedingly difficult to obtain (Phillips 1999). The above discussion suggests that (1) most of the work done for fixing 'critical threshold values' for external influences on catastrophic slope instability events (e.g. Dikau & Schrott 1999; R y b ~ 1999, 2004; Schmidt & Dikau 2004) is biased by methodical incorrectness; (2) there is a theoretically given limit for adequacy of results from linearly-causally based numerical models of slope deformation behaviour and stability failure, which cannot be overcome by any further refinement (even for the most sophisticated ones, such as FEM and DEM methods). On the other hand,

NONLINEAR SCIENCE ISSUES IN ROCK SLOPES there is also an exponential law stating the minimal amount of data needed for plausible conclusions about time-series patterns in nonlinear analyses. The practical limit to the embedding dimension of time series that can be analysed in practice has been fixed at a value of 4 (e.g. Hunt et al. 2003). Therefore, when analysing relatively short time series (as is quite normal in many practical cases) rather dangerous assumptions have to be adopted, either that the dynamics of the variable chosen for that analysis is affected by only a few state variables (e.g. Henttonen & Hanski 2000) or that such a small embedding level allows detection of nonlinearity s.s. even for large-dimensional time series (e.g. Nychka et al. 1992). In any case, applying inadequately modified methods to data of poor quality or insufficient quantity may result in mathematically inconsistent or implausible findings; hence, a multidisciplinary approach is inevitably necessary for elaborating appropriate models in the spatio-temporal domain. A possible way of simulating the process of rock slope collapse preparation as the development of hierarchically structured, complex systems with multifactor control is by using Self Organized Criticality models, which qualitatively differ from the currently used engineering-oriented ones. In this way, quite new pieces of knowledge could be revealed that have not yet been discovered through spatially-temporally limited field observations (see Holland 1998; Bossomaier & Green 2000).

(3)

(4)

91

distinguishing between intrinsic slope movement dynamics and climatically driven SA activity. The qualitative difference between intrinsic slope movement dynamics of the NTE and FFE time series is important for assessing slope behaviour. The NTE series possess a linear non-Gaussian but asymmetrical fattailed distribution of movement events. In contrast, the FFE series are nonlinear (s.1.) features of persistent long-range correlation of self-affine processes with two scaling regions. The graphical methods and the numerical testing of fractal features seem to be very promising for assessing the state of immediate rock fall danger. To this end, it also suggested that modelling the dynamics of preparation for rock slope collapse as a complex self-organizing system may be appropriate to reveal the crucial dynamical patterns of slope failure systems.

The study was supported by T110190504 Project of the Academy of Sciences of Czech Republic, by MSN 00216 20831 Project of Ministry of Education, Youth and Sports of Czech Republic, and the Institutional Research Plan AVOZ10300504. The authors wish to thank the editors for their valuable comments, stimulating questions, and manuscript corrections, which helped to improve the overall quality of the paper.

Conclusions

References

The main conclusions arrived at with this study are listed as follows:

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(1)

(2)

Monitoring time series from unstable rock slopes reveals more information about dynamics of slope failure than we are able to acquire by current analytical methods. The reason for this is that current methods are based on a linearly causal paradigm, and are founded on empirical-phenomenological models, which are unable to be sufficiently assisted by the processing power of computers when dealing with large amounts of data. In the case history reported here, the hidden information was revealed when methods from a toolbox of the Nonlinear Sciences were applied. Geometrical methods, the phase portrait, and correlogram of time series have proven to possess greater ability to display the features looked for than the current time-displacement plots. In particular, they proved to be more suitable for revealing transitions from near-to-equilibrium (NTE) to far-from-equilibrium (FFE) dynamical states of slope failure system, as well as to help in

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Multifractal variability in self-potential signals measured in seismic areas L U C I A N O T E L E S C A 1, V I N C E N Z O L A P E N N A t & M A R I A M A C C H I A T O 2

1Institute of Methodologies for Environmental Analysis, CNR, C.da S.Loja, 85050 Tito (PZ), Italy (e-mail: [email protected]) 2Dipartimento di Scienze Fisiche, INFM, Universitgt 'Federico H', Naples, Italy Abstract: Multifractal variability in the time dynamics of geoelectrical data, recorded in a

seismic area of southern Italy, was studied by means of Multifractal Detrended Fluctuation Analysis (MF-DFA), which allows the detection of multifractality in nonstationary signals. Our findings show that the multifractality of the geoelectrical time series recorded in the study area is mainly due to the different long-range correlations for small and large fluctuations. Furthermore, the singularity spectrum has led to a better description of the signal, revealing a clear enhancement of its degree of multifractality (measured by the variation of the standard deviation of the generalized Hurst exponents h(q) or the Hrlder exponents t~) in association with the occurrence of the largest earthquake. However, in order to assess significant correlations between large earthquakes and patterns of multifractal parameters, an investigation of data sets covering longer periods and different seismotectonic environments is needed. The study also furnishes details on our approach for investigating the complex dynamics of earthquake-related geoelectrical signals.

Research devoted to investigating earthquakes and earthquake-related geophysical variability has been the subject of growing interest in recent years. Monitoring the time variability of several geophysical parameters may be useful in understanding phenomena linked to seismic activity (Rikitake 1988; Zhao & Qian 1994; Park 1997; Martinelli & Albarello 1997; Di Bello et al. 1998; Vallianatos & Tzanis 1999; Hayakawa et al. 2000; Telesca et al. 2001; Tramutoli et al. 2001). In particular, variations in the stress and fluid flow fields can produce changes in the self-potential field (Scholz 1990), which may be used to obtain information on the governing mechanisms both in normal conditions and during intense seismic activity. Self-potentials are voltage differences between two points on the Earth's surface. These are caused by the presence of an electric field produced by natural sources distributed in the subsoil (e.g. Parasnis 1986; Sharrna 1997; and references therein). The most relevant phenomenon that may originate self-potential anomalous fields is known as streaming potential; the electrical signal is produced when a fluid flows in a porous rock, due to a pore pressure gradient. The phenomenon is generated by the formation within the porous ducts of a double electrical layer between the bounds of the solid, which absorbs electrolytic anions and cations distributed in a diffused layer near the boundaries. The dissolved salts increase the amount of anions and cations of the underground liquids. The free liquid in the centre of the rock

pore is usually enriched and cations, and anions are usually absorbed on the soil surface in silicate rock. The free pore water carries an excess positive charge, a part of which accumulates close to the solid-liquid interface forming a stable double layer. When the liquid is forced through the porous medium, the water molecules carry free positive ions in the diffusion part of the pore. This relative movement of cations with reference to the firmly attached anions generates the well-known streaming potential (Keller & Frischknecht 1966), which, as suggested by Mizutani et al. (1976), may be responsible for the voltage pattern detected at the ground surface before a major earthquake (Patella 1997). In a seismic focal region, the effect could be enhanced due to increasing accumulation of strain, which can cause rock dilatancy (Nur 1972). Self-potential dynamics could reflect the irregularity and heterogeneity of the crust, within which phenomena generating self-potential fields occur. Therefore, the structure of the self-potential signal could be linked to the structure of the seismic focal zone. In fact, the geometry and the structure of individual fault zones can be represented by a network with an anisotropic distribution of fracture orientations, and consisting of fault-related structures including small faults, fractures, veins and folds. This is a consequence of the roughness of the boundaries between each component and the interaction between the distinct components within the fault zone (O'Brien et al. 2003). In

From: CELLO,G. & MALAMUD,B. D. (eds) 2006. Fractal Analysisfor Natural Hazards. Geological Society, London, Special Publications, 261, 95-103. 0305-8719/06/$15.00 9 The Geological Society of London 2006.

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L. TELESCA ETAL.

fact, earthquake faulting is characterized by irregular rupture propagation and non-uniform distributions of rupture velocity, stress drop and coseismic slip. These observations indicate a nonuniform distribution of strengths in the fault zone, whose geometry and mechanical heterogeneity are important factors to be considered in the prediction of strong motion. Experimental studies on the hierarchical nature of the processes underlying fault rupture, leading to the possibility of recognizing the final preparation stage before a large earthquake occurs, have been performed by Lei et al. (2003), whereas Cowie et al. (1993) introduced a numerical rupture model to simulate the growth of faults in a tectonic plate driven by a constant plate boundary velocity. They found that the plate initially deforms by uncorrelated nucleation of small faults reflecting the distribution of material properties. Because of the increase of strain, the growth and coalescence of existing faults dominate over nucleation, and a power-law distribution of fault size appears, characterizing the fault pattern as fractal. Cowie et al. (1995) also show that the combined effect of fault clustering and the correlation between fault displacement and fault size leads to a strongly multifractal deformation pattern. To characterize quantitatively self-potential (SP) dynamics, techniques able to extract robust features hidden in their complex fluctuations are needed. Fractality is one of the features of such complexity. A fractal is an object whose sample path that is included within some radius r scales with r. Fractal processes are characterized by scaling behaviour, which leads naturally to power-law statistics. Consider f(x), which depends continuously on the scale x over which the measurements are taken. Suppose that changing the scale x by a factor a will effectively scale the statistics f ( x ) by another factor g(a), f ( a x ) = g(a)f(x). The only non-trivial solution for this scaling equation is given byf(x) = bg(x), g(x) = x c, for some constants b and c (Thurner et al. 1997 and references therein). Therefore, power-law statistics and fractals are very closely related concepts. The fractality of a signal can be investigated with the aim of characterizing its temporal fluctuations; in this case, we need to perform second-order fractal measurements, which furnish information regarding the correlation properties of a time series. Spectral analysis represents one of the standard methods to detect correlation features in time series fluctuations. The power spectrum is obtained by means of the Fourier transform of the signal. It describes how the power is concentrated at various frequency bands. Thus, the power spectrum reveals periodic, multiperiodic, or non-periodic signals. The fractality of a time series is revealed by a power-law dependence of the spectrum upon

the frequency, S ( f ) ~ l / f , where the scaling (spectral) exponent ce yields information on the type and the strength of the time-correlation structures intrinsic in the signal fluctuations (Havlin et al. 1999). If ot = 0, the temporal fluctuations are purely random, typical of white noise processes, characterized by completely uncorrelated samples. If ot > 0, the temporal fluctuations are persistent, meaning that variations of the signal will be very likely followed by positive (negative) variations; this feature is typical of systems that are governed by positive feedback mechanisms. If a < 0, the temporal fluctuations are antipersistent, meaning that variations of the signal will be very likely followed by negative (positive) variations; this feature is typical of systems that are governed by negative feedback mechanisms. The estimate of the spectral exponent is rather rough, as a result of large fluctuations in the power spectrum, especially at high frequencies. Furthermore, the power spectrum is sensitive to nonstationarities that could be present in observational data. For this reason, different fractal methods, such as the Higuchi method or detrended fluctuation analysis, have been developed to furnish stable estimates of the spectral exponent (Higuchi 1988, 1990), or to allow the detection of scaling behaviours in experimental time series, very often affected by trends and nonstationafities, which cause spurious detection of correlations (Peng et al. 1995). All the above techniques are monofractal, and very often they are not sufficient to describe the scaling properties of a signal, that could be associated with a multifractal object (i.e. an object that needs many exponents to characterize its scaling properties). In this case, the signal can be decomposed into many subsets characterized by different scaling exponents. Thus, multifractals are intrinsically more complex and inhomogeneous than monofractals, and characterize systems featuring irregular dynamics, with sudden bursts of highfrequency fluctuations. The aim of the present paper is the dynamical investigation of a self-potential time series recorded in southern Italy, one of the most seismically active areas in the Mediterranean Region. Our purpose is to characterize the multifractality of such time series in order to reveal a possible connection with the seismic activity of the area.

Data We studied a geoelectrical data set recorded at the Giuliano station (40.688~ 15.789~ located in one of the most seismically active areas in southern Italy (Fig. 1). The signal consists of voltage difference between two non-polarizable electrodes

MULTIFRACTALITY IN SELF-POTENTIAL SIGNALS 1~ ~

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Fig. 1. Location of the Giuliano geoelectrical station in southern Italy. This station has been installed in one of the most seismically active areas of the Mediterranean Region, struck by strong earthquakes in past (1857) and recent years (1980).

inserted 1 m deep in the ground to avoid external temperature effects. The distance between the electrodes was 100 m. Figure 2 shows a time series, which consists of minute-sampled geoelectrical values, recorded from 1 March 2001 to 31 May 2003, and the location of the earthquakes with magnitude M > 3.0 that occurred in the area during the observation period. These values satisfy Dobrovolsky's rule (Dobrovolsky et aL 1979, 1993), which is a theoretical relation between earthquake magnitude, distance from the epicentre, and volumetric strain, and states that detectable seismically induced strain exceeds 10 -8 . From this relation the maximum distance from the epicentre in which the effects of the earthquake are detectable is r = 100"43M, where r is measured in km. It may be observed, from Figure 2, that the signal shows large fluctuations in association with the occurrence of

the selected earthquakes; these are sharper for the largest event (M = 4.1) of the seismic sequence.

Methods

and

data

analysis

Observational data often show clear irregular dynamics, characterized by sudden bursts of highfrequency fluctuations. This suggests performing a multifractal analysis in order to detect the possible occurrence of different scaling behaviours for different intensities of fluctuations. Furthermore, the signal may often appear to be nonstationary. Multifractal Detrended Fluctuation Analysis (MF-DFA) (Kantelhardt et al. 2002) is a useful tool to characterize multifractality in nonstationary data. The method is based on the conventional detrended fluctuation analysis (Peng et al. 1995).

L. TELESCA ETAL.

98

for v = Ns + 1. . . . . 2Ns. Here, y~(i) is the fitting line in segment v. Then, an average over all segments is performed to obtain the qth order fluctuation function

100-

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o.'o '~.ox~e',.ox~o ~'~.oX~o~'~.oX,o~~.o~,e'~,~o~ t(Min) (1 March 2001 to 31 May 2003)

Fig. 2. Time variation of the geoelectrical signal, measured at the Giuliano site, and the earthquakes (vertical arrows) that occurred in the area and satisfy Dobrovolsky's rule (Telesca et al. 2004). The signal consists of minute-sampled voltage differences between two non-polarizable electrodes inserted 1 m deep in the ground to avoid external temperature effects. The distance between the electrodes is 100 m and the data were recorded from 1 March 2001 to 31 May 2003.

It operates on the time series x(i), where i = 1, 2 . . . . . N and N is the length of the series. The mean value is indicated by Xave. Assuming that x(i) are increments of a random walk process around the average Xaw, the 'trajectory' or 'profile' is given by the integration of the signal i

y(i) = Z Ix(k) - Xave].

(1)

k=l

Next, the integrated time series is divided into

Ns = int(N/s) non-overlapping segments of equal length s. Because the length N of the series is often not a multiple of the considered time-scale s, a short part at the end of the profile y(i) may remain. In order not to disregard this part of the series, the same procedure is repeated starting from the opposite end. Therefore, 2Ns segments are obtained in total. The local trend for each of the 2Ns segments is then calculated by a least-square fit of the series. Then one calculates the variance

F2(s, v) =

s 1 ~__2_ {y[(v -

1)s + i] - yv(i)}2

(2)

S i=1

for each segment v, v ---- 1. . . . . Ns and

FZ(s,v )

[

1_~ {y[N - (v -Ns)s+ i] -yv(i)} 2 - - S i=1

(3)

where, in general, the index variable q can assume any real value except zero. Repeating the procedure described above, for several time-scales s, Fq(s) will increase with increasing s. Then, analysing l o g - l o g plots Fq(s) v. s for each value of q, the scaling behaviour of the fluctuation functions can be determined. If the series xi is long-range power-law correlated, Fq(s) increases for large values of s as a power-law Fq(S) OC.Sh(q).

(5)

The value h(0) corresponds to the limit h(q) for q--+ 0, and cannot be determined directly using the averaging procedure of Eq. (4) because of the diverging exponent. Instead, a logarithmic averaging procedure has to be employed,

1 2us Fo(s) -- exP l-~S~=lln[F2(s,v)]

} ,~ s h(O). (6)

In general, the exponent h(q) will depend on q. For stationary time series, h(2) is the well-defined Hurst exponent H (Feder 1988). Thus, we call h(q) the generalized Hurst exponent. Monofractal time series are characterized by h(q) independent of q. The different scaling of small and large fluctuations will yield a significant dependence of h(q) on q. For positive q, the segments v with large variance (i.e. large deviation from the corresponding fit) will dominate the average Fq(s). Therefore, if q is positive, h(q) describes the scaling behaviour of the segments with large fluctuations, and generally, large fluctuations are characterized by a smaller scaling exponent h(q) for multifractal time series. For negative q, the segments v with small variance will dominate the average Fq(s). Thus, for negative q values, the scaling exponent h(q) describes the scaling behaviour of segments with small fluctuations, usually characterized by larger scaling exponents. Two types of multifractality that underlie the q-dependence of the generalized Hurst exponent in time series can be discriminated: (1) due to a broad probability density function for the values of the time series, and (2) due to different long-range

MULTIFRACTALITY IN SELF-POTENTIAL SIGNALS

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correlations for small and large fluctuations. Both of them need a multitude of scaling exponents for small and large fluctuations. The easiest way to discriminate between these two types of multifractality is by analysing the corresponding randomly shuffled series. In the shuffling procedure the values are put into random order and, although all correlations are destroyed, the probability density function remains unchanged. Hence, the shuffled series coming from multifractals of type (2) will exhibit simple random behaviour with hshuf(q)=0.5, which corresponds to purely random dynamics. Instead, those coming from multifractals of type (1) will show h(q) = hshuf(q), because the multifractality depends on the probability density. If both types of multifractality characterize the time series, the shuffled series will show weaker multifractality than the original one. Figure 3 provides three fluctuation functions Fq(s) (q = - 10, 0, 10) for the self-potential signal measured at the Giuliano station for time-scales s ranging from 5 • 102 min to N/4, where N is the total length of the series. The length of the series (N ~ 1.2 • 106) allows us to consider the estimated exponents reliable. The fluctuation functions present different slopes, estimated by a leastsquare method (the coefficient of correlation is 0.99 in all three cases); this suggests the presence of multifractality in the series. Figure 4 shows the q-dependence of the generalized Hurst exponent h(q) determined by fits in the regime 5 • 102 rain < s < N / 4 and for q ranging between - 1 0 and 10 with 0.5 steps. The generalized Hurst exponents v. q, averaged over 10 randomly shuffled versions of the original time series are also shown. The error bars delimit the 1 - tr range around the mean values. The hshuf(q) values range around 0.5, but with a slight q-dependence; this indicates that the

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Fig. 4. The h(q)-q relation for the original and shuffled series. The difference between the two h(q) spectra is very clear, suggesting that the multifractality of the original signal is significant and depends on the different long-range correlations between small and large fluctuations.

multifractality of the self-potential data is mainly due to the different long-range correlations for small and large fluctuations. The multifractal scaling exponents h(q) defined in Eq. (6) are directly related to the scaling exponents 7(q) defined by the standard partition function multifractal formalism (Kantelhardt et al. 2002). Suppose that the series xk is a stationary, positive and normalized sequence. Then the detrending procedure of the MF-DFA is not required. Thus the DFA can be replaced by the 'fluctuation analysis' (FA), for which the variance is defined as F2A(S, V) = [y(vs) -- y((v -- 1)S)]2.

(7)

Inserting this definition into Eq. (4) and using Eq. (6), we obtain 1

ly(vs) - y((v - 1)s)l q

~ s h(q).

(8)

v=l

For the sake of simplicity, we assume that the length N of the sequence is an integer multiple of the scale s, obtaining Ns = N / s and therefore Ns Z [y(vs) --y((v -- 1)S)Iq , ~

s qh(q)-I

.

(9)

v=l

The term [ y ( v s ) - y ( ( v - 1 ) s ) ] is the sum of the numbers xk within each segment v of size s. This sum is known as the box probability ps(v) in the standard multifractal formalism for normalized series xk.

L. TELESCA ETAL.

100

The scaling exponent ~-(q) is usually defined via the partition function Zq(s),

9 original shuffled 1.0 o

Ns Zq(s) = ~ Ip,(v)l q ~ s "r(q)

(1o)

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where q is a real parameter as in the MF-DFA. Equation (10) is identical to Eq. (9), therefore

0.4-

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(1 l)

0.2-

In this equation, h(q) is different from the generalized multifractal dimensions D(q) = ~ ( q ) / ( q - 1); in fact, although h(q) is independent of q for a monofractal series with compact support, D(q) depends on q in that case (Kantelhardt et al. 2002). The assumption of compact support of the series leads to the fractal dimension of the support D(0) = - T(0) = 1. Therefore, monofractal series with long-range correlations are characterized by the linearly dependent q-order exponent z(q), that is, the exponents r(q) of different moments q are linearly dependent on q

0.0-

r(q) = Hq -- 1

(12)

with a single Hurst exponent,

H = dr/dq = const.

(13)

Long-range correlated multifractal signals have a multiple Hurst exponent, that is, the generalized Hurst exponent h(q),

h(q) = d~'/dq :~ const

/

-0.8 ' -0'.6 '-0'.4 ' -0'.2

| 0'.0'0:2'0'.4

0'.6'

a

Fig. 5. Singularity spectra of the original and shuffled series. The spectrum of the original series is clearly wider than that of the shuffled series, suggesting a higher multifractality degree.

We investigated the time variation of the multifractal behaviour of the series in order to find possible correlations with the local earthquakes occurring during the observation period. We calculated the set of the generalized Hurst exponent {hq(t):-lO < q < +10} in overlapping time windows 10 5 minutes long; the time shift between two successive windows was set to 5 • 10 3 min. Figure 6 shows the time variation of the h(q) spectrum; strong variability is clearly visible. Figure 7 shows the time variation of the multifractal

(14)

where -r(q) depends nonlinearly on q (Ashkenazy et al. 2003). The singularity spectrum f ( a ) is related to z(q) by means of the Legendre transform (Parisi & Frisch 1985), OT a ----- dq

f ( a ) = qa - "r(q)

(15) (16)

where a is the Hrlder exponent and f ( a ) indicates the dimension of the subset of the series that is characterized by a. The singularity spectrum quantifies in detail the long-range correlation properties of a time series. Figure 5 shows the multifractal spectrum f(a) for the original and the shuffled series. The spectrum of the original series is completely different from that of the shuffled series, indicating a significant multifractality.

Fig. 6. Time variation of the h(q)-q spectrum. The generalized Hurst exponents {hq(t): -10 < q < +10} are calculated in overlapping time windows l0 5 min long; the time shift between two successive windows was set to 5 x 1 0 3 min. The strong variability of the h(q) spectrum with time indicates an analogous variability of the multifractality of the signal.

MULTIFRACTALITY IN SELF-POTENTIAL SIGNALS

Fig. 7. Time variation of the singularity spectmmf(a). The spectra are obtained from the Legendre transform applied to overlapping time windows l0 5 min long; the time shift between two successive windows was set to 5 • l0 3 min. The width of the singularity spectrum changes with time, indicating a significant variability of the multifractality of the signal.

101

Fig. 9. Time variation of the H/ilder a value. A clear enhancement of the range of the a values in association with the occurrence of the largest earthquake (M = 4.1) of the recorded seismic sequence is visible.

spectrum, obtained by the L e g e n d r e transform f r o m

the h(q) spectrum. The multifractal spectrum also presents a strong variability with time, m o r e visible in the projection on the plane f ( a ) - a (Fig. 8), w h i c h shows a clear e n h a n c e m e n t o f the range o f the a values in association with the occurrence o f the largest earthquake (Fig. 9). Figures 10 and 11 show the standard deviation o f the h(q) exponents and the a values varying with time (both parameters can be considered a m e a s u r e o f the degree o f multifractality o f the series); clear e n h a n c e m e n t of both parameters is visible in correspondence with the largest earthquake.

Fig. 8. f(a)-a projection of the graph plotted in Figure 7. This projection clearly shows the variability of the width of the singularity spectrum with time.

Fig. 10. Time variation of the standard deviation of the h(q) exponents. This parameter can be considered as a measure of the multifractality degree and shows a clear enhancement in correspondence with the largest earthquake.

Fig. 11. Time variation of the standard deviation of the Hrlder a value. This parameter can be considered as a measure of the multifractality degree and shows a clear enhancement in correspondence with the largest earthquake.

102

L. TELESCA ETAL.

Conclusions The geophysical phenomenon underlying the selfpotential variability connected to earthquake activity is complex and is governed by physical laws that are not completely known. Multifractal analysis has led to a better understanding of such complexity, by means of the generalized Hurst exponents and the singularity spectrum. Using the surrogate series, obtained by a random shuffling procedure, the multifractality of the series is shown to be mainly due to different long-range correlations for small and large fluctuations. The singularity spectrum has led to a better description of the signal, revealing a clear enhancement of its degree of multifractality (measured by the variation of the standard deviation of the generalized Hurst exponents h(q) or the Hrlder exponents a) in association with the occurrence of the largest earthquake. In order to assess significant correlations between large earthquakes and patterns of multifractal parameters, and to became more confident in using such patterns to perform feasible earthquake prediction, we need however to investigate data sets covering longer periods, and different seismotectonic environments. The use of multifractal tools is nevertheless promising for better characterizing the time dynamics of earthquake-related geophysical phenomena.

References ASHKENAZY, Y., HAVLIN, S., IVANOV, P. CH., PENG, C.-K., SCHULTE-FROHLINDE, V. & STANLEY, H. E. 2003. Magnitude and sign scaling in powerlaw correlated time series. Physica A, 323, 19-41. COWIE, P. A., VANNESTE, C. t~; SORNETTE, D. 1993. Statistical physics model for the spatio-temporal evolution of faults. Journal of Geophysical Research, 98, 21809-21821. COWIE, P. A., SORNETTE, D. & VANNESTE, C. 1995. Multifractal scaling properties of a growing fault population. Geophysical Journal International, 122, 457-469. DI BELLO, G., HEINICKE,J., KOCH, U., LAPENNA, V., MACCHIATO, M., MARTINELLI, G. 8z PISCITELLI, S. 1998. Geophysical and geochemical parameters jointly monitored in a seismic area of Southern Apennines (Italy). Physics and Chemistry of the Earth, 23, 909-914. DOBROVOLSKY,I. P. 1993. Analysis of preparation of a strong tectonic earthquake. Physics of the Solid Earth, 28, 481-492. DOBROVOLSKY,I. P., ZUBKOV,S. I. & MIACHKIN,V. I. 1979. Estimation of the size of earthquake preparation zones. Pageoph, 117, 1025-1044. EEDER, J. 1988. Fractals. Plenum Press, New York. HAVLIN, S., AMARAL, L. A. N., ASHKENAZY, Y., GOLDBERGER, A. L., IVANOV,P. CH. PENG, C.-K. & STANLEY, H. E. 1999. Application of statistical

physics to heartbeat diagnosis. Physica A, 274, 99-110. HAYAKAWA, M., HATTORI, K., ITOH, T. & YUMOTO, K. 2000. ULF electromagnetic precursors for an earthquake at Biak, Indonesia on February 17, 1996. Geophysical Research Letters, 27, 15311534. HIGUCHI, T. 1988. Approach to an irregular time series on the basis of the fractal theory. Physica D, 31, 277 -283. HIGUCHI, T. 1990. Relationship between the fractal dimension and the power law index for a time series: a numerical investigation. Physica D, 46, 254-264. KANTELHARDT, J. W., ZSCHIEGNER, S. A., KONSCIENLY-BUNDE, E., HAVLIN, S., BUNDE, A. & STANLEY, H. E. 2002. Multifractal detrended fluctuation analysis of nonstationary time series. Physica A, 316, 87-114. KELLER, G. V. & FRISCHKNECHT, F. C. 1966. Electrical Methods in Geophysical Prospecting. Pergamon Press, Oxford. LEI, X., KUSUNOSE, K., NISHIZAWA, O. & SATOH, T. 2003. The hierarchical rupture process of a fault: an experimental study. Physics of the Earth and Planetary Interiors, 137, 213-228. MARTINELLI, G. & ALBARELLO, D. 1997. Main constraints for siting monitoring networks devoted to the study of earthquake related hydrogeochemical phenomena in Italy. Annali di Geofisica, 40, 1505-1522. MIZUTANI, H., ISHIDO,T., YOKOKURA,T. & OHNISHI, S. 1976. Electrokinetic phenomena associated with earthquakes. Geophysical Research Letters, 3, 365-368. NUR, A. 1972. Dilatancy pore fluids and premonitory variations of ts/tp travel times. Bulletin of the Sebmological Society of America, 62, 1217-1222. O'BRIEN, G. S., BEAN, C. J. & MCDERMOTT,F. 2003. A numerical study of passive transport through fault zones. Earth and Planetary Science Letters, 214, 633-643. PARASmS, D. S. 1986. Principles of Applied Geophysics, Chapman and Hall, London. PARISI, G. & FRISCH, U. 1985. A multifractal model of intermittency. In: GHIL, M., BENZI, R. & PARISl, G. (eds) Turbulence and Predictability in Geo-

physical Fluid Dynamics and Climate Dynamics. North Holland, Amsterdam, 84-88. PARK, S. K. 1997. Monitoring resistivity changes in Parkfield, California 1988-1995. Journal of Geophysical Research, 102, 24545-24559. PATELLA, D. 1997. Introduction to ground surface SP tomography. Geophysical Prospecting, 45, 653-681. PENG, C.-K., HAVLIN, S., STANLEY, H. E. & GOLDBERGER, A. L. 1995. Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos, 5, 82-87. RIKITAKE, T. 1988. Earthquake prediction: an empirical approach. Tectonophysics, 148, 195-210.

MULTIFRACTALITY IN SELF-POTENTIAL SIGNALS SCHOLZ, C. H. 1990. The Mechanics of Earthquakes and Faulting. Cambridge University Press, New York. SHARMA, P. S. 1997. Enviromental and Engineering Geophysics. Cambridge University Press, Cambridge. TELESCA,L., CUOMO, V., LAPENNA,V. 8z MACCHIATO, M. 2001. A new approach to investigate the correlation between geoelectrical time fluctuations and earthquakes in a seismic area of southern Italy. Geophysical Research Letters, 28, 4375-4378. TELESCA, L., COLANGELO, G., LAPENNA, V. ~z MACCHIATO, M. 2004. Fluctuation dynamics in geoelectrical data: an investigation by using multifractal detrended fluctuation analysis. Physics Letters A, 332, 398-404.

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THURNER, S., LOWEN, S. B., FEURSTEIN, M. C., HENEGHAN,C., FEICHTINGER,H. C. & TEICH, M. C. 1997. Analysis, synthesis, and estimation of fractal-rate stochastic point processes. Fractals, 5, 565 -596. TRAMUTOLI, V., DI BELLO, G., PERGOLA, N. & PISCITELLI, S. 2001. Robust satellite techniques for remote sensing of seismically active areas. Annali di Geofisica, 44, 295-312. VALLIANATOS, F. & TZANIS, A. 1999. On possible scaling laws between Electric Earthquake Precursors (EEP) and Earthquake Magnitude. Geophysical Research Letters, 26, 2013-2016. ZHAO, Y. & QIAN, F. 1994. Geoelectric precursors to strong earthquakes in China. Tectonophysics, 233, 99-113.

A general landslide distribution applied to a small inventory in Todi, Italy D O N A L D L. T U R C O T T E l, B R U C E D. M A L A M U D 2, F A U S T O G U Z Z E T T I 3 & PAOLA REICHENBACH 3

1Department of Geology, University of California, Davis, CA, 95616, USA (e-mail: turcotte@ geology.ucdavis.edu) 2Environmental Monitoring and Modelling Research Group, Department of Geography, King's College London, Strand, London WC2R 2LS, UK (e-mail: [email protected]) 3CNR-IRPI Perugia, via della Madonna Alta 126, Perugia 06128, Italy (e-mail: fausto, guzzetti @irpi. cnr. it; paola, reichenbach @irpi. cnr. it) Abstract: Large numbers of landslides can be associated with a trigger, for example, an earthquake or a large storm. We have previously hypothesized that the frequency-area statistics of landslides triggered in an event are well approximated by a three-parameter inverse-gamma distribution, irrespective of the trigger type. The use of this general distribution was established using three substantially complete and well-documented landslide event inventories: 11,000 landslides triggered by the Northridge California Earthquake, 4000 landslides triggered by rapidly melting snow cover in the Umbria region of Italy, and 9000 landslides triggered by heavy rainfall associated with Hurricane Mitch in Guatemala. In this paper, we examine further this general landslide distribution by using an inventory of 165 landslides triggered by heavy rainfall in the region of Todi, Central Italy. Our previous studies have shown the applicability of our general landslide distribution to events with 4000-11,000 landslides. This smaller inventory provides a critical step in examining the applicability of the general landslide distribution. We find very good agreement of the Todi event with our general distribution. This also provides support for our further hypothesis that the mean area of landslides triggered by an event is approximately independent of the event size.

Landslides are complex phenomena influenced by many factors, including soil and rock types, bedding planes, topography, and moisture content. Landslide events consist of one to many thousands of landslides, generally associated with a trigger such as an earthquake, a large storm, a rapid snowmelt, or a volcanic eruption. A landslide event may be quantified by the frequency-area distribution of the triggered landslides. We have recently shown (Malamud et al. 2004a) that the f r e q u e n c y - a r e a statistics of three substantially complete landslide inventories are well approximated by the same probability density function, a three-parameter inverse-gamma distribution. We also introduced a landslide-event magnitude scale mL = 1og(NLT), with NET the total number of landslides associated with the landslide event, in analogy to the Richter earthquake magnitude scale. We argue that the statistics of triggered-landslide events under a wide variety of conditions follow the same general statistical behaviour to a good approximation. Such a 'general' statistical behaviour is widely accepted for the frequency-magnitude statistics of earthquakes, which are also complex, but generally follow a power-law relationship

between the number of earthquakes and the earthquake rupture area, the Gutenberg-Richter relation. In this paper, we will first discuss our 'general' probability distribution of landslide areas for triggered landslide events, and then several implications of having a 'general' distribution, including (1) a magnitude scale for landslide events, (2) the same theoretical average area of landslides associated with any given landslide event, and (3) the ability to extrapolate for incomplete landslide events or historical inventories. We will also present a fourth, much smaller, landslide inventory from Todi, central Italy.

Frequency-area distributions In order to give the statistical distribution of landslide areas, a probability density function p(AL) is defined according to 1 tSNL p(AL) ----NET tSAL

From: CELLO,G. & MALAMUD,B. D. (eds) 2006. FractalAnalysisfor NaturalHazards. Geological Society, London, Special Publications, 261, 105-111. 0305-8719/06/$15.00 9 The Geological Society of London 2006.

(1)

106

D.L. TURCOTTE E T A L .

with the normalization condition AL)dAL = 1

landslide size) and then decreasing with a powerlaw tail. The inventories were estimated to be nearly complete (Harp & Jibson 1995, 1996; Cardinali et aL 2000; Bucknam et al. 2001; Guzzetti et al. 2002) for landslides with length scales greater than 5 - 1 5 m (AL ~ 25-225 me), therefore the 'rollover' in Figure 1 is regarded as real. Based on the good agreement between these three sets of probability densities, we proposed (Malamud et al. 2004a) a general probability distribution for landslides, a three-parameter inverse-gamma distribution, given by (Johnson & Kotz 1970; Evans et aL 2000)

(2)

where AL is landslide area, NLT is the total number of landslides in the inventory, and 8NL is the number of landslides with areas between AL and AL + 8AL. In Figure 1 we present the probability densities p(AL) for three substantially complete landslide inventories, from the USA, Italy and Guatemala. A detailed discussion of each inventory is found in Malamud et al. (2004a). The three sets of probability densities given in Figure 1 exhibit a characteristic shape (Guzzetti et al. 2002; Malamud et al. 2004a), with densities increasing to a maximum value (most abundant

p(AL; P, a, s ) =

exp -(3)

L a n d s l i d e area, A L ( m z)

101 103

102

102 i

i

i i iii

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~

105 i

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[

10-3

i i i ill

[] Northridge earthquake

6

%~

o Umbria snowmelt

~

, Guatel2al2 r2i2~all

u

10-4

104

10-6 ~,

~, 10~

8001

,~ lO~

10_2

10-3

i

10-7

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600

10-8

400ti

~

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~\

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10-9

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i iiii1

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- 10_11

L a n d s l i d e area, A L (knl 2)

Fig. 1. Dependence of landslide probability densities p on landslide area AL, for three landslide inventories (figure after Malamud et al. 2004a): (1) 11,111 landslides triggered by the 17 January 1994 Northridge earthquake in California (Harp & Jibson 1995, 1996); (2) 4233 landslides triggered by rapid snowmelt in the Umbria region of Italy in January 1997 (Cardinali et al. 2000; Guzzetti et al. 2002); (3) 9594 landslides triggered by heavy rainfall from Hurricane Mitch in Guatemala in late October and early November 1998 (Bucknam et al. 2001). Probability densities are given on logarithmic axes in (a) and linear axes in (b). Also included is our proposed general landslide probability distribution. This is the best-fit to the three landslide inventories of the three-parameter inverse-gamma distribution of Eq. (3), with p = 1.40, a -----1.28 • 10 -3 km2, and s = 1.32 • 10-4 km2 (coefficient of determination r 2 = 0.965).

A GENERAL LANDSLIDE DISTRIBUTION where F(O) is the gamma function of p. The inversegamma distribution has a power-law decay with exponent - ( p + 1) for medium and large areas and an exponential rollover for small areas. The maximum likelihood fit of Eq. (3) to the three data sets in Figure 1 yields p = 1.40, a = 1.28 x 10 -3 km 2, and s = 1.32 x 10 -4 km 2, with coefficient of determination r 2 = 0.965; the power-law tail has exponent p + 1 = 2.40. Many authors (see Malamud et al. 2004a for a review) have also noted that the freq u e n c y - a r e a distributions of large landslides correlate with a power-law tail. This common behaviour is observed despite large differences in landslide types, topography, soil types, and triggering mechanisms. On the basis of the good agreement between the three landslide inventories and the inverse-gamma distribution illustrated in Figure 1, Malamud et al. (2004a) hypothesized that the distribution given in Eq. (3) is general for landslide events. It is not expected that all landslide-event inventories will be in as good agreement as the three considered, but we do argue that the quantification, if only approximate, is valuable in assessing the landslide hazard (Guzzetti et al. 2005, 2006).

107

In this paper we present the probability densities p(AL) for a fourth landslide inventory consisting of NLT = 165 rainfall-triggered landslides in the vicinity of Todi, central Italy, with landslides occurring in the period March to May 2004. The inventory was compiled through reconnaissance field surveys, and is reasonably complete. Probability densities for these landslides are given in Figure 2, along with the inverse-gamma distribution from Eq. (3) with p = 1.40, a = 1.28 x 10 -3 km 2, and s = 1.32 x 10 -4 km 2, the best-fit to the three inventories in Figure 1. A reasonable agreement is obtained between this fourth set of data (Fig. 2) from Todi, Italy, and the inverse-gamma distribution. It should be emphasized that the total number of landslides in the Todi event, NLT = 165, is a factor of thirty to eighty less than the number of landslides in the three substantially complete inventories given in Figure 1. Before discussing implications of a 'general' landslide distribution, we briefly discuss rockfall inventories. It has been shown (Dussauge et al. 2003; Malamud et al. 2004a) that the f r e q u e n c y size statistics of rockfalls are very different than the statistics for other types of landslides as

Landslide Area, A L (m 2) 101

102

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Landslide Area, A L (km 2) Fig. 2. Dependence of landslide probability densities p on landslide area AL, for 165 rainfall-triggered landslides in the vicinity of Todi, central Italy, occurring March to May 2004. Probability densities are given on logarithmic axes. Also included is the three-parameter inverse-gamma distribution of Eq. (3) with p = 1.40, a = 1.28 • 10 - 3 km 2, and s ---- 1.32 x 10 - 4 k m 2, the best-fit to the three inventories in Figure 1.

108

D.L. TURCOTTE ETAL.

discussed above. Malamud et al. (2004a) examined three rockfall inventories, and found that the noncumulative f r e q u e n c y - v o l u m e relationship is best described by a power-law with exponent -1.93, with no 'rollover' for smaller rockfall volumes; the equivalent tail for the medium and large landslides in our 'general' distribution Eq. (3) with p = 1.40, with areas converted to volumes, would be a power-law exponent of - 1 . 0 7 . There is a significant difference in slope for the medium and large landslides, compared to rockfalls, and the rockfalls have no 'rollover' for the smaller landslides. This difference has been attributed to different applicable physics, with rockfalls controlled by processes of fragmentation, compared to landslides that are primarily controlled by the process of slope stability. We now discuss the implication of our general landslide distribution for the 'average' landslide area in the landslide event.

Average landslide area Assuming the validity of Eq. (3) for the probability distribution of landslide areas in individual triggered landslide events, Malamud et al. (2004a) used the distribution to derive a theoretical mean landslide area AL. This is the mean of all landslide areas that occur during a landslide event. The theoretical mean area is obtained by taking the first moment of the probability distribution function, giving AL =

ALp(AL)dAc.

(4)

Substitution of the three-parameter inverse-gamma distribution from Eq. (3) into Eq. (4) and integrating gives AI~ = - -a p-1

+ s.

(5)

For the landslide probability distribution given in Figure 1 we have p = 1.40, a = 1.28x 10 - 3 k m 2, and s = 1.32 x 10 4 km 2, so that AL = 3.07 x 10 .3 km 2. One implication of our landslide distribution is that because the probability distribution always has the same mean, all landslide events should have the same mean landslide area AL = 3.07 x 10 .3 k i l l 2 = 3070 m 2. This follows directly from the applicability of our proposed landslide distribution, and is independent of the number of landslides associated with a landslide event. Malamud et al. (2004a) found that the measured mean landslide areas AL for the three event inventories, Northridge, Umbria, and Guatemala, are A L = 3 . 0 1 x 10- 3 k i n2, 2 . 1 4 x 10 - 3 k m 2, and 3.07 x 10 3km2, in good agreement with the value predicted by our general landslide

distribution. The mean landslide area for the fourth inventory from Todi, Italy, given in Figure 2, is AL = 4.05 x 10 -3 km 2. This is also in reasonably good agreement with the theoretical AT. = 3.07 x 10 -3 k m 2, particularly for a triggered landslide event with so few landslides, and is potential further confirmation of our general landslide distribution.

Landslide magnitude scale A second implication of having a 'general' landslide probability distribution is the ability to create a landslide event magnitude scale. Measures of event sizes are useful for natural hazards. For example, the Richter magnitude scale for earthquakes is universally used and the general public has some understanding of the implications of an M = 7.0 earthquake. Over a dozen magnitude scales are available for other natural hazards, including the Saffir-Simpson scale (hurricanes), the Fujita scale (tornadoes), and the Volcanic Explosivity Index. Malamud et al. (2004a) proposed a magnitude scale mi. for a landslide event based on the logarithm to the base 10 of the total number of landslides associated with the landslide event: mL = log NLT.

(6)

Keefer (1984) and later Rodr/guez et al. (1999) used a similar scale to quantify the number of landslides in earthquake-triggered landslide events: 100-1000 landslides were classified as a two, 1000-10,000 landslides a three, and so on. The landslide event magnitudes for the three event inventories considered by Malamud et al. (2004a) are (1) Northridge earthquake-triggered event, mc = 4.05; (2) Umbria snowmelt-triggered event, mL = 3.63; (3) Guatemala rainfall-triggered event, mL = 3.98. These are in the range of mr_ = 3.6-4.0. Although observed earthquakes span a wide range on the Richter scale, available landslideevent inventories are often restricted to a relatively narrow magnitude range. There are several reasons for this. Accurate inventories are restricted to populated areas and have been carried out only during the last ten years or so. Thus, very few large landslide events with mc > 4.0 have occurred in regions where studies have been carried out. In addition, there has been little incentive to carry out studies of small-magnitude landslide events, mc < 3.6. A limited range was also the case for earthquakes in the 1940s, when instrumental limitations limited studies to large earthquakes and the timespan for studies was short, so few large earthquakes had occurred. Therefore, one of the purposes of this paper was to introduce a fourth

A GENERAL LANDSLIDE DISTRIBUTION 'substantially' complete landslide inventory, but with a low magnitude. The Todi rainfall-triggered event has mE ----2.22. Given several hundred landslide events and their magnitudes in a given region and time period, we hypothesize that there will be many more 'smaller' magnitude events compared to the larger ones. In analogy to earthquakes, we further hypothesize that these will follow a relationship such that log Arc = --bmL + a, where Nc is the number of landslide events with magnitudes greater than or equal to mL, and b and a are constants. To partially confirm (or disprove) this hypothesis, we will need to assign landslide event magnitudes using substantially complete inventories, or making extrapolations based on 'incomplete' inventories, which we now discuss.

Historical and incomplete inventories An historical landslide inventory includes the sum of many landslide events that have occurred over time. Assuming that our landslide probability distribution is applicable to all landslide events, the sum of events over time (the historical inventory) will also satisfy this distribution (Malamud et al. 2004a). However, in historical inventories, the evidence for the existence of many smaller and medium landslides will have been lost due to wasting processes over time. Therefore, for the historical inventories, we attribute the deviation from our general landslide distribution to the incompleteness of the inventories. Using the general landslide distribution of Eq. (3), we can extrapolate an inventory that contains just the largest landslides to give the total number and total volume of all landslides in the region. Malamud et al. (2004b) used this extrapolation by considering two examples. Frequency densities were used because the inventories are incomplete and the normalization given in Eq. (2) no longer holds. From Eq. (1), the frequency densityf(AL) is /SNL f(AL) --- NLTp(AL). 3AL

(7)

Theoretical curves of f(AL) for various landslideevent magnitudes mL are obtained by multiplying the probability distribution p(AL) given in Eq. (3) by the total number of landslides in the event NET. Curves are given in Figure 3 for mE = 1 (NLT = 10) to mL = 8 (NLT = 108). The same method can be used for a single incomplete landslide event inventory where only the largest landslides have been measured (e.g. those largest landslides that comprise just 1-2% of the total inventory) and the medium and smaller

109

landslide sizes are not known. Using the general landslide distribution Eq. (3), we can extrapolate the frequency densities of the largest landslides to give the total number of all landslides in the event, and estimate the equivalent landslide event magnitude. Figure 3 includes frequency densities for two historical landslide inventories, from Italy and Japan. Also included, for reference, are frequency densities for the snowmelt-triggered Umbria landslide event (Fig. 1). The first historical inventory includes 44,724 landslides in Umbria, Italy (Guzzetti et al. 2003), estimated to have occurred in the last 5 10 ka (thousand years). The power-law tail of the frequency densities is in good agreement with the landslide-event probability distribution of Eq. (3). With a landslide magnitude of mL = 5.8 _ 0.1, we estimate that over the last 5 - 1 0 ka the total number of landslides that have occurred is NET = 650,000 __ 150,000. The second historical inventory in Figure 3 includes 3424 landslides in the Akaishi Ranges of central Japan (Ohmori & Sugai 1995) estimated to have occurred in the last 10 ka. The power-law tail of the frequency densities gives mL : 6.0 ___0.2, corresponding to NET : 1,100,000 ! 500,000. Malamud et al. (2004b) related the landslideevent magnitude for individual events to the total area and volume of associated landslides, as well as the area and volume of the maximum landslides. They then used the historical landslide inventories just discussed (Fig. 3) from Italy and Japan, and made estimates of total area and volumes of landslides involved over time for each of the regions, and from these a lower bound estimate on regional erosion rates due to landslides. They inferred longterm erosion rates due to landslides in these two regions of Italy and Japan as 0.4 and 2.2 mm year- 1, respectively.

Conclusions Landslide events display large variations in landslide types, sizes, distributions, patterns, and triggering mechanisms. Many would question whether such complex phenomena can be quantified. Malamud et al. (2004a) showed that three well-documented and substantially complete landslide-event inventories from different parts of the world, each with different triggering mechanisms, have frequencyarea statistics that are well approximated by a threeparameter inverse-gamma distribution (Eq. 3). In this paper, we have shown that a fourth, much smaller landslide inventory, is also well-approximated by our proposed 'general' landslide distribution. It is clearly desirable to test this distribution using other substantially complete landslides inventories.

110

D.L. TURCOTTE ETAL. Landslide area, A L (m 2)

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Fig. 3. Dependence of the landslide frequency densityfon landslide areaAL, both on logarithmic axes (after Malamud et al. 2004a). Landslide frequency distributions corresponding to our proposed landslide probability distribution (Eqs. 3 and 7) are given for landslide magnitudes mL : 1, 2 . . . . . 8 (NET = 10, 102. . . . . 108). Also included are the frequency densities for three landslide inventories: (1) Umbria snowmelt-tfiggered landslides as in Figure 1; (2) 3424 historical landslides in the Akaishi Ranges of central Japan (Ohmori & Sugai 1995) estimated to have occurred in the last 10 ka; (3) 44,724 historical landslides in the Umbria region of Italy (Guzzetti et al. 2003) estimated to have occurred in the last 5-10 ka.

There are several important implications of the applicability of a general landslide distribution. It provides the basis for defining a landslide-event magnitude scale me = log(NLX), with NEW the total number of landslides in the landslide event. The direct determination of the landslide-event magnitude requires that the landslide inventory be substantially complete. However, the general landslide distribution can be used to determine a landslideevent magnitude from a partial inventory, where the inventory is complete only for landslide areas greater than a specified value. It can also be used for historical inventories, which include the sum of landslide events over time. Using the general landslide distribution, the total area and volume of associated landslides in the event or sum of events over time, as well as the area and volume of the

maximum landslides, can be directly related to the landslide-event magnitude mL. These can then be used to estimate regional erosion due to landslides. The contributions of author D.L.T were partially supported by NSF Grant No. ATM 0327558 and the contributions of authors B.D.M., F.G., and P.R. were partially supported by the European Commission's Project No. 12975 (NEST) 'Extreme events: Causes and consequences (E2-C2)'. References

BUCKNAM, R. C., COE, J. A. ETAL. 2001. Landslides triggered by Hurricane Mitch in Guatemala inventory and discussion. U.S. Geological Survey Open File Report, 01-443.

A GENERAL LANDSLIDE DISTRIBUTION CARDINALI, M., ARDIZZONE, F., GALLI, M., GUZZETTI, F. & REICHENBACH,P. 2000. Landslides triggered by rapid snow melting: The December 1996-January 1997 event in Central Italy. In: CLAPS, P. & SICCARDI, F. (eds) Proceedings 1st Plinius Conference on Mediterranean Storms, Maratea, Italy, 14-16 October 1999, Consenza: Editorial BIOS, 439-448. DUSSAUGE, C., GRASSO, J.-R. & HELMSTETTER, A. 2003. Statistical analysis of rockfall volume distributions: Implications for rockfall dynamics. Journal of Geophysical Research, 108, 2286 (11 p.), doi: 10.1029/2001JB000650. EVANS, M., HASTINGS, N. & PEACOCK, J. B. 2000. Statistical Distributions, 3rd edn. John Wiley, New York. GUZZETTI, F., MALAMUD, B. D., TURCOTTE, D. L. & REICHENBACH, P. 2002. Power-law correlations of landslide areas in central Italy. Earth and Planetary Science Letters, 195, 169-183. GUZZETTI, F., REICHENBACH, P., CARDINALI, M., ARDIZZONE, F. & GALLI, M. 2003. Impact of landslides in the Umbria Region, Central Italy. Natural Hazards and Earth System Sciences, 3, 469-486. SRef-ID: 1684-9981/nhess/2003-3-469. GUZZETTI, F., REICHENBACH, P., CARDINALI, M., GALLI, M. & ARDIZZONE, F. 2005. Probabilistic landslide hazard assessment at the basin scale. Geomorphology, 72, 272-299. GUZZETTI, F., GALLI, M., REICHENBACH, P., ARDIZZONE, F. & CARDINALI,M. 2006. Landslide hazard assessment in the Collazzone area, Umbria, central Italy. Natural Hazards and Earth System

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Sciences, 6, 115-131, SRef-ID: 1684-9981/ nhess/2006-6-115. HARP, E. L. & JIBSON, R. L. 1995. Inventory of landslides triggered by the 1994 Northridge, California earthquake. U.S. Geological Survey Open File Report, 95-213. HARP, E. L. & JIBSON, R. L. 1996. Landslides triggered by the 1994 Northridge, California earthquake. Seismological Society of America Bulletin, 86, $319-$332. JOHNSON, N. L. & KOTZ, S. 1970. Continuous Univariate Distribution. John Wiley, New York. KEEFER, D. K. 1984. Landslides caused by earthquakes. Geological Society of America Bulletin, 95, 406-421. MALAMUD, B. D., TURCOTTE, D. L., GUZZETTI, F. & REICHENBACH, P. 2004a. Landslide inventories and their statistical properties. Earth Surface Processes and Landforms, 29, 687-711, doi: 10.1002/esp. 1064. MALAMUD, B. D., TURCOTTE, D. L., GUZZETTI, F. & REICHENBACH, P. 2004b. Landslides, earthquakes, and erosion. Earth and Planetary Science Letters, 229, 45-59. OHMORI, H. & SUGAI, T. 1995. Toward geomorphometric models for estimating landslide dynamics and forecasting landslide occurrence in Japanese mountains. Zeiscrift fur Geomorphology (Suppl. Bd.), 101, 149-164. RODRIGUEZ, C. E., BOMMER, J. J. & CHANDLER, R. J. 1999. Earthquake-induced landslides: 1980-1997. Soil Dynamics and Earthquake Engineering, 18, 325 -346.

Scaling properties of the dimensional and spatial characteristics of fault and fracture systems in the Majella Mountain, central Italy L. M A R C H E G I A N I a, J. P. V A N D I J K 2, P. A. G I L L E S P I E 3, E. T O N D I 1 & G. C E L L O 1

1Dipartimento di Scienze della Terra, Universitgt di Camerino, Via Gentile III da Varano, 62032 Camerino (MC), Italy (e-mail: [email protected]; emanuele.tondi @unicam, it; giuseppe.cello@ unicam.it) 2Eni S.p.A. E&P Division, Via Emilia 1, 20097 San Donato Milanese (Mi), Italy (e-mail: Janpieter. [email protected]) 3Norsk Hydro, Oil and Energy Research Centre, Bergen, N-5020 Bergen, Norway Abstract: In this paper we report on the results of a systematic study carried out on the fault and

fracture systems exposed in the Majella Mountain, in the central Apennines fold and thrust belt of mainland Italy. The focus of our work was to assess the dimensional, spatial, and scaling properties of fault and fractures in carbonate rocks, in order to set up appropriate flow models for these types of potential geofluid reservoirs. The results provide information on (1) orientation, size distribution, density variations, and fractal characteristics of the fault and fracture networks affecting the Majella anticline; (2) the scaling properties and the overall architecture of different fault zone components; (3) the overprinting relationships between fault and fracture sets and the Majella fold structure. These data were used to elaborate a three-dimensional discrete fault and fracture model (DFFN model) of a ~100 m3 geological volume, and for this to (1) evaluate the transport and storage properties of the reservoir; and (2) assess the degree of vulnerability and any possible hazard related to the exploitation and management of geofluids hosted in carbonate rock volumes.

The Majella Mountain (Fig. 1) was selected for this study because it represents a good analogue for fractured geofluid reservoirs (i.e. deep aquifers, oil, gas, and geothermal fields). In particular, the Majella Mountain is thought to be a possible analogue for the oil-producing subsurface structures discovered a few years ago in southern Italy (Lampert et al. 1997). Furthermore, excellent exposures of different types of structural discontinuities in carbonate rocks, and the occurrence of an outcropping Lower Cretaceous to Pliocene multilayered sequence, furnish a good opportunity to collect a detailed and homogeneous data set on fault and fracture networks affecting various lithological units at different scales (from microcracks to regional scale features). Multiscale fault and fracture networks dissecting the various tectonic units of the Apennines represent the most relevant features allowing geofluid migration or storage within subsurface structures, depending on their permeability and on the sealing capacity of major fault zones. The basic concepts and methods needed to evaluate the different factors and components controlling fault seal behaviour have been reviewed by Moller-Pedersen & Koestler (1997). According to these authors, the most important steps in fault seal analysis include (1) the definition of the geometry, space

distribution, and orientation of major fault zones; (2) the assessment of the architecture and related permeability structure of major fault zones, and the density distribution of lower-rank (i.e. subseismic) faults and fractures; and (3) the evaluation of the sealing mechanisms and critical juxtapositions representing compartment boundaries. In fractured reservoirs, the largest faults are usually mappable by using regional geological and seismic data, whereas smaller faults and fractures are detected in wells. However, data from wells can only provide very limited information about fracture size, clustering, aperture, and connectivity, which all together determine their flow properties. For this reason, detailed studies on field analogues of fractured reservoirs are required to collect additional information for (1) predicting fault and fracture distributions at the subseismic scale, and (2) working out realistic flow models. In the last two decades, it has been shown that faulting is a self-organized process on geological timescales (Bak & Tang 1989; Cowie et al. 1993), as many attributes of natural fault networks (including their dimensional and spatial properties) show power-law (fractal) distributions (Walsh & Watterson 1988; Marret & Allmendinger 1991; Gillespie et al. 1992; Barton 1995; Schlische et al. 1996; Cello, 1997; Cowie 1998; Cowie &

From: CELLO, G. & MALAMUD,B. D. (eds) 2006. FractalAnalysisfor Natural Hazards. Geological Society, London, Special Publications, 261, 113-131. 0305-8719/06/$15.00 9 The Geological Society of London 2006.

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Fig. 1. (a) Geological sketch map of the Majella area, and location of the measuring sites. (b) Simplified geological cross-section (modified from Scisciani et al. 2002).

FRACTURE SYSTEMS IN CENTRAL ITALY Roberts 2001; Tondi & Cello 2003). Consequently, points (1) and (2) above may both benefit from a detailed fractal analysis of fault and fracture distributions, as well as of their spatial properties. In this study, a large amount of data on exposed fault and fracture networks was collected in order to assess their dimensional and spatial properties as well as the possible relations to the main regional structures (i.e. major faults and macrofolds). In particular, we investigated in detail (1) the relations between fracture density and folds in carbonate rocks (Hennings et al. 2000; Johnson & Johnson 2000), (2) the spacing, length, and displacement distributions of both faults and fractures (see Caine et al. 1996; Yielding et al. 1996; Cello et al. 2000; among others), (3) the scaling properties and spatial distribution of fault-related structures (see Gillespie et al. 1993, 2001, among others), (4) the fault and fracture length distributions within the same geological structure (see Van Dijk et al. 2000, among others). The above information provided the appropriate input data needed for setting a discrete fault and fracture model matching well test data, and for evaluating the sealing potential of the major outcropping fault zones.

Geological setting The Majella Mountain is part of the external zone of the central Apennines, in the Abrnzzo region of central Italy (refer to Fig. 1). The stratigraphic succession includes a 2-km-thick sequence of carbonate rocks belonging to the most external units of the Apennines (Eberli et al. 1993; Lampert et al. 1997). The sequence records the sedimentary conditions of a complex platform to basin depositional system that developed in the area from Middle Jurassic times, when the pre-existing carbonate platform of the Apulian domain was bounded by northward dipping normal faults (Eberli et al. 1993). Since the Early Cretaceous, this fault system produced a large marginal escarpment separating the southern carbonate platform from the northern pelagic environment, where a stratigraphic sequence including carbonate grainstones was deposited. During the Neogene, the overall succession was covered by carbonate clastics and evaporitic deposits interbedded within a siliciclastic foredeep sequence. From a structural viewpoint, the investigated area is formed by an upthrust asymmetric box-shape anticline; in plain view, the fold shows an arcuate shape with an axis curving from N W - S E (in the northern sector) to N E - S W in the southern one. This structure, which developed mostly at the end of the Early Pliocene, is bounded along its

115

western side by a major fault system (the Caramanico fault system, CFS) displaying about 4 km of cumulative displacement (Ghisetti & Vezzani 2002). The CFS separates the carbonate units of the Majella structure from Early Pliocene clastic units (e.g. the Majella Flysch) interposed between the Morrone units (belonging to a more internal carbonate platform domain) and the Majella Mountain. Along the eastern limb of the Majella fold there occur (1) an unconformable contact between the underlying carbonate units and the external Pliocene-Early Pleistocene terrigenous deposits cropping out in the northern sector, and (2) a back-thrust superposing the Majella units above the clastic and siliciclastic allochthonous deposits cropping out in the central and southern sectors of the anticline. Several strike-slip faults and a few N W - S E and east dipping late normal faults (with maximum displacements of a few hundred metres) cut through the northern sectors of the area, whereas transpressional and rotated normal faults occur mainly along the eastern flank of the anticline. The latter may represent the result of reactivation, during compression, of the early normal faults (Scisciani et al. 2002).

Systematic structural analysis The geo-structural data collected in the Majella area include regional, stratigraphic and tectonic information, as well as a systematic database on the spatial and dimensional properties of the outcropping fault and fracture networks. The above data were acquired by means of area sampling (scan area analysis), linear sampling (scan line analysis) and volumetric sampling (performed by means of photogrammetric surveys of the exposed fault and fracture populations). Scan area data (including fractures types, orientation, aperture, connectivity, and two-dimensional spatial patterns) were acquired mostly within a few square metres wide zones in the protolith, whereas scan line data about fracture properties, as well as on their spacing and directional characteristics, were collected along transects across the main faults (see Fig. 4); photogrammetric surveys were instead used to collect fault and fracture data from inaccessible outcrops (i.e. steep mountain fronts, deep canyons, and so on). Scan area analysis was carried out at 62 sites, mainly in areas located between fault-bounded blocks, in order to provide information on the fractured protolith. Scan line analysis was instead performed along 40 transects cutting through a few main fault zones. Fault-related fracture

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parameters were measured along a reference line perpendicular to the strike of the fault; in this way we were able to assess fracture orientation, spacing, aperture, and frequency distribution within the fault zone, and to estimate the relative size of each fault component (core and damage zone). The results of scan line analysis proved to be appropriate for the assessment of the permeability structure of selected rock volumes. Volumetric sampling was performed by means of three-dimensional stereophotogrammetric surveys at four selected sites in the northern sector of the Majella anticline (in the S. Spirito gorge, near Fara S. Martino; along the Pennapiedimonte river; in the Madonna della Mazza and Pretoro quarries; see Fig. 1 'key sites' for locations). The stereophotogrammetric surveys allowed us to obtain digital photos from which we performed a threedimensional analysis of the imaged faults and fractures. Most of the photographs were shot from a distance of 2 - 5 0 m, and their positioning was based on high-precision triangulations connected to the National Topographic Network of Italy. The resulting photographic frame was then imported on a PC-based system and provided a highprecision Digital Terrain Model of the outcrop. The above data allowed us to recognize different kinds of structural discontinuities, including faults, hybrid fractures, stylolites, and joints; the faults often contain breccia and gouge and show slikensided surfaces, whereas open joints are marked by a characteristic plumose structure. The main properties of the analysed features are illustrated below. Faults

The majority of faults exposed in the Majella area are characterized by normal or strike-slip

kinematics; visible displacements usually range from a few centimetres to about 200 m, with the exception of the Caramanico fault system, occurring along the western fold limb, which displays cumulative dip-slip (normal) displacements of a few thousand metres (Ghisetti & Vezzani 2002). Distinctive fault types, in the study area, were related to the mechanical properties of the different lithological units and to their timing of activity. We distinguished the following features. Pre-tilting faults. These affect the lithotypes belonging to both platform margin and basinal sequences. Where possible, they were ascribed, according to their characteristic features, either to pre- or to syn-diagenetic deformation events (Marchegiani et al. 1999). Systematic tilt correction (Fig. 2) turned out to be useful in recognizing these faults, especially where no other information was available. Faults in porous grainstones. These often occur as fault clusters within the Cretaceous grainstones of the Orfento Fm. Each fault zone includes fault rocks formed by crushed grains cemented by micrite; in general, these fault zones display a reduced porosity. Fault zone thickness is quite variable, and the overall geometry of the clustered faults (Fig. 3) is similar to that described as deformation bands in porous sandstones (see Antonellini & Aydin 1995). Faults in massive limestones with calcarenites. These are discrete faults that may be analysed in the lithotypes of the Tre Grotte Fm., Scaglia Fm. and Cima delle Murelle Fm. The fault damage zone and the fault core of these structures generally display a thickness ranging from tens of centimetres to a few millimetres. Faults in bioclastic limestones. Localized fault zones embedded within a roughly non-fractured protolith are characteristic of the Bolognano Fm.

Fig. 2. (a) Present-day orientation of pre-tilting faults (in the Cima delle Murelle Fm., Fara S. Martino area); (b) tilt-corrected orientations. Please, note that in (b) there is much less dispersion of fault orientation data.

FRACTURE SYSTEMS IN CENTRAL ITALY

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Fig. 3. Deformation bands in the grainstones of the Orfento Fm. (Madonna della Mazza quarry).

These fault zones include many fault-related slip surfaces and different fault rock types (mostly gouge and microbreccia). In the following section we present a few diagrams, selected from our database, which are representative of the different attributes of the analysed fault and fracture sets, in order to discuss the main points that emerged from this study. Figure 4 plots various classes of orientations derived from scan line analysis. In this type of representation, the spatial position of each fracture is plotted as a function of the distance from the scan line origin (i.e. from where the fault intersects the scan line); as a result, one can estimate the contribution of each discontinuity set to the overall fracture frequency distribution along the transect. Our database on fracture frequency also shows that in general, in most fault zones, one or two highfrequency peaks may be recognized, and that these control the spatial characteristics of the fracture mesh making up the damage zone (Cello et al. 2000, 2001) as well as the permeability structure of the faults (see also Caine et al. 1996; Evans et al. 1997; Sibson 2000). We also observed that, in the fault damage zone, fracture frequency increases two to three times compared to the 'background' frequency, which is almost constant, as it records regional features within the protolith. In order to compare different fault zones, the cumulative spacing distribution of fault-related fractures was normalized with reference to the scale of observation; that is, the cumulative number of

fractures was divided by the length (L) of each sample line, and then reported in l o g - l o g plots like the one shown in Figure 5. As may be observed from the resulting diagram, the relation between spacing (S) and weighted cumulative number of fractures (N) can be fitted by a power law of the form N ( > S) _ cS_ D L

(1)

where D = - 0.93. The validity range of the above relation is within at least four orders of magnitude. In Figure 6, typical frequency distributions of fault-related fractures affecting different lithological units are shown. The distribution associated with faults dissecting the Tre Grotte Fm. (a) shows a gradual increase of the fracture number as the main fault is approached, whereas the Bolognano Fm. (b) is characterized by a peak with a constant highfrequency value. Fractures related to fault zones in the Orfento Fm. (c) are often clustered and 'fill up' the whole sample line with highfrequency values ( > 3 % ) and a few peaks (>6%). The pre-tilting faults (d), cutting mostly through Cretaceous limestones, display well-identified peaks, which, however, are not always related to distinct faults (d) and often define fracture swarms. Fault spacing was analysed in detail at M. della Mazza quarry (see Fig. 1 for location), where more than 120 NNE and SSW

F a u l t spacing.

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Fig. 4. Frequency v. distance diagram from Passo Lanciano section. The five 3D lines from front to back are the classes of orientation, 30/40 ~ to 260/270 ~. Note that only two of the five fracture sets shown in the diagram display an increase in frequency with respect to the 'background frequency', hence suggesting that these two sets are fault-related, whereas the other three include fractures inherited from previous deformation events recorded by the protolith.

dipping meso-faults displacing the Cretaceous grainstones of the Orfento formation are nicely exposed. Here, we also computed the coefficient of variation (Cv) of the spacing distribution, which is the standard deviation of the spaces divided by the m e a n spacing (Gillespie et al. 1993), as this furnishes additional information on fault spatial properties. Cv is in fact greater than 1 for faults that are clustered, less than 1 for faults that are regularly spaced, and equal to 1 for faults that are random. In the Madonna della Mazza area, the Cv coefficient for the NNE dipping faults is 1.04, and that of the

SSW dipping faults is - 0 . 9 6 ; these values suggest therefore that their distribution is almost random. In order to investigate the spatial distribution of the mesofaults further, adjacent faults of the two sets were plotted on a cumulative frequency diagram with l o g - l i n e a r axes (Fig. 7). As can be seen, for both sets, the data points lie approximately on a straight line, hence indicating a negative exponential cumulative frequency distribution. This distribution is consistent with a random distribution of the faults, and so it is in agreement with the results of Cv analysis.

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Fault displacement. Fault displacement distribution was analysed systematically by measuring fault throw along sample lines in the Madonna della Mazza quarry, and across the Pennapiedimonte section (see Fig. 1 for locations). Additional data from larger faults were derived from specifically constructed geological cross-sections across the Vallone di S. Spirito and from available geological sections (Donzelli 1997). The results of this analysis are summarized in Figure 8. As may be seen, the point distribution relative to displacements ranging from 1 cm to 100 m is scale invariant and can be fitted by a power law with a relatively high value ( - 0 . 8 5 ) of the power-law exponent, hence suggesting that most of the displacement is accommodated by small fault segments. Displacement-length relations were derived mainly from small faults characterized by displacements of between 2 and 30 cm and lengths of less than 600 cm. As may be seen in Figure 9a, there exists a linear relation between these two fault parameters (see also Walsh & Watterson 1993). The same data set shows that the size of the fault damage zone (Ft) is also linearly related to the displacement (Fig. 9b). On the other hand, the geometry of fault displacement profiles, which altogether are predominantly symmetric (see for example Fig. 10), suggests that the analysed mesofaults were mostly generated by the radial growth of isolated non-interacting structures (see also Cello 2000; Scholz & Gupta 2000).

Fractures Different types of fractures have been recognized in the Majella area; they have been grouped according to their kinematics, shape, aperture, filling, and so on. Most of the analysed fractures, which are characterized by plumose markings and subplanar geometries, were recorded as joints. The majority of joints are barren, but in a few cases we observed different types of mineralization, including: (1) (2) (3)

Fe, Mn deposits; oil (in the Lettomanoppello and Decontra quarries); calcite (in the Pretoro quarry).

Stylolites ('anti-cracks' of Fletcher & Pollard 1981) are also present in the strata of the Majella stratigraphic succession. They occur mainly as bed-parallel features, and are best interpreted as pre-tilting structures (see also Graham et al. 2003). Stylolites oblique to bedding are also present, and these have been related to slip along bedding surfaces and/or to strike-slip displacement across transpressional faults (as in the Pennapiedimonte valley). The different types of fractures mapped in the Majella area may be designated in three main sets (Fig. 11). (1)

Set L This is a roughly E - W trending set, which is predominant in the central and southern sectors of the fold. Thin section analysis shows that some of the fractures belonging to this set are sediment-filled

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........................................

-~

i

"

" "','~.

0,001

,~

.......

y=o.o?ssx-~

0.0001

~^ i

0.001

0.01

;

0.I

i

~ I0

i "u I00

121

I000

The existence of pre-tilting joints in the central sector of the Majella Mountain (i.e. where outcropping rocks belong mostly to the platform margin area) is also suggested by the results of fracture restoration by bedding rotation, which shows a strong clustering of the joints along an E - W trend (refer to Fig. 11). The relative timing among the three main sets was established by taking into account their cross-cutting relationships; these show that set I is the oldest, whereas set II and Ill developed almost at the same time during folding. The latter two sets have also been locally reactivated in recent times (i.e. in the current stress regime). This observation, which is mostly based on detailed meso-strucrural analyses carried out in the Fara S. Martino area, indicates that features belonging to set I were reactivated as left lateral shears (Fig. 12), whereas the N E - S W features are coherently arranged within a right lateral shear zone. Moreover, both sets cut the bed-parallel stylolitic surfaces. The dimensional and spatial properties of the outcropping fractures were assessed by analysing the cumulative frequency distribution of features such as width, length, spacing, density and connectivity (Gillespie et al. 1993; Yielding et al. 1996; Van Dijk et al. 2000; Ortega & Marret 2000). Fracture width was estimated by measuring their apertures at different structural positions, and within various lithological units of the sedimentary sequence involved in the Majella anticline. The results of this analysis (Fig. 13) show that a major peak in the weighted frequency distribution occurs along the N N W - S S E trend. In our opinion, this may be related to the opening effect induced by the regional stress field acting in the area over the last 700,000 years (Cello et al. 1997; Di Bucci & Mazzoli 2002; Mariucci & Mfiller 2003; Goes et al. 2004). Fracture length was derived from cumulative frequency plots such as those shown in Figure 14. As can be seen, the relationship governing fracture length distribution at different scales may be expressed by a power law of the form:

Displacement (m)

Fig. 8. Scaled cumulative fault displacement distribution. Displacement data were obtained from faults sampled at the Pennapiedimonte and Madonna della Mazza areas, and from available geological sections (from Donzelli 1997). Note that the distribution may be fitted by a power-law relation that is characterized by a negative exponent (D = -0.85).

N( > L) _ cL_ D

(2)

A where A is the value of the reference surface, c is a constant and D is the power-law exponent. In the Majella Mountain, the computed c value is 1.3, and the power-law exponent is -1.88. In

122

L. MARCHEGIANI ET AL.

(a)

35 y = 0.046x ~" 3O - - R 2 = 0.69 f A

'~ 9 25

E ~ 20

Q. ~ 15 I~

~ 10

4'I~ ~ ~O~ '~I~

i

i

i

i

i

100

200

300

400

500

600

L, Fault length (cm)

(b) 6o

9

Ft= 0.70 D so

R 2 = 0.71 t

40 ..~

"

~

30

9

~ o

.~

~ ~

.I~. ~

~176176149

14.

~ 2o

~ o~ ~176 .~

-'6

0

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~

9

i

i

J

t

J

i

10

20

30

40

50

60

D, Displacement (cm) Fig. 9. (a) Fault displacement-length and (b) fault zone thickness-displacement diagrams relative to small faults cropping out at Madonna della Mazza and Pennapiedimonte.

Figure 14a, the length value was obtained from scan areas analysis covering two orders of magnitude; this was compared with the cumulative frequency distribution of fractures mapped by means of photogrammetric analysis (Fig. 14b), and with the cumulative distribution of fault length derived from the Majella structural map and from the analysis of the photogrammetric images of the Madonna della Mazza quarry (Fig. 14c). All the data points shown in Figure 14 are related by the power-law expression D = - 1 . 8 8 . Our results therefore suggest that, despite the lack of data in the range 100 m to 1 km, the fracture and fault length distribution, in the study area, are scale invariant over at least five orders of magnitude.

Fracture spacing within the protolith was obtained by constructing fracture logs along strata within different lithological units (Fig. 15). This type of data is available for the central sector of the Majella Mountain and concerns mostly Cretaceous limestones belonging to the basinal sequence. (Tre Grotte Fm., Orfento Fm., S. Spirito Fm. and Bolognano Fro.) The results of our work are expressed quantitatively by the mean spacing value and by the coefficient of variation (Cv). From fracture log analysis we estimated Cv values ranging from 0.52 to 0.76. These values indicate a relatively regular frequency distribution. The computed mean spacing value ranges from 1 to 18 cm. As shown in Figure 16, this variability is sometimes

FRACTURE SYSTEMS IN CENTRAL ITALY 14

D =-0.O013L

123

2 + 0.3398L o

....12

t:::

~

121

2 0 0

50

100

150

200

250

L, Length (cm) Fig. 10. Fault displacement profile (from the Pennapiedimonte area).

Fig. 11. Fracture orientation data relative to the Majella structure as a whole. Note that the rose diagrams highlight the occurrence of three main fracture sets, which display a better definition after bed tilt corrections.

124

L. MARCHEGIANI E T AL.

Fig. 12. Re-sheared N 170 ~ E oriented joints affecting the Cima della Murelle Fm. at the Fara S. Martino gorge.

Fig. 13. Open fractures distribution as a function of their orientation. The NNW-SSE oriented cluster (vertical dashed line) is interpreted as being a result of the opening effect of the currently active stress field.

FRACTURE SYSTEMS IN CENTRAL ITALY

125

(a) 1000 ~ ....

y = 1.3 x -1"8857

II ~11,I

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o : Lo,;]~

[] S A 3 a SA4

..o

10

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E

o SA7

Z o SA9 a SA 10

E

0.1

SAIl I+ S A 1 2

0

0.001 0.001

0,1 Length (m)

(b) 1000 y = 0.21 x -18z75

I...

_&

lO

..D

E

o Fara 3 § Fara 2 * Fara 4 Fara 5 x Fara 6

0.1

o Fara 7 9 .

E O

'~.~

o Fara 32 9 + Fara27

0.001

0.00001 0.00001

0.001

0.1

10

1000

Length(m)

(c) 100000

I t 8837 y = 0.33 x-

~,

o,.7

9

.11

E

9

,~

I~ Majelta faults

]

o.1

Z

.~

o.ool

:\~\

=

E

0.00001

",, \

0.0000001

,~

0.000000001 1E-09

1E-07

0.00001

0.001

0.1

10

1000

100000

Length(m)

Fig. 14. Scaled fracture length distributions derived from (a) scan areas, (b) photogrammetric data, and (c) geological map data sets. The distributions may all be fitted by a power-law relation covering about five orders of magnitude.

126

L. MARCHEGIANI ET AL.

Fig. 15. Fracture logs and Cv values within single beds from the Pennapiedimonte section.

strongly dependent upon bed thickness (B), and is clearly expressed, in the range of 5 cm to 1 m, by the relation (Fig. 16) S ~ B~

(3)

with a spacing-to-layer-thickness ratio S / B ~ 0.13. Fracture density was estimated from scan line and scan area data. In the first case, we obtained fracture density values conventionally expressed as number of joints per sample line length (P10; Hennings et al. 2000): PIO ~--

1

S(avg)

(4)

where S(avg ) is the average value of spacing. In the second case, we derived two-dimensional fracture density values conventionally expressed as length of joints per sample area (P21): P21 - )--~L A

(5)

where L is the fracture trace length and A the size of the scan area. In order to minimize the influence of lithology on fracture density, we used a 'correction parameter' 6 to obtain a lithology-weighted density value. The

parameter 6 is expressed as 6 -- P21 (avg) P21 (avg)

(6)

where P21(avg) is the average fracture density value, and P21~avg) is the average P21 value for a given formation. As shown in Figure 17, P21 values are highly variable, depending on structural location within the Majella fold. It may be seen, in fact, that the spatial distribution of P21 displays higher values along the eastern limb of the anticline and low values in the area between the fold hinges. Fracture connectivity is a structural parameter of paramount importance for assessing the fluid flow/ storage properties of fractured rocks. It may be estimated from two-dimensional data that allow one to discriminate between (1) isolated (type I), (2) coupled (type II), and (3) interconnected (type III) fractures (Ortega & Marret 2000). The input data for assessing fracture connectivities in the study area were provided as digitized maps obtained from scan areas and photogrammetric data. The results of this type of analysis are shown in Figure 18. As can be seen from the triangular diagram of Figure 18, the fracture connectivities in the Tre Grotte Fm. and the Bolognano Fm. are quite different. The Tre Grotte Fm. is mostly characterized by interconnected

FRACTURE SYSTEMS IN CENTRAL ITALY

127

100.

S~-B

0.55

A

E 0

01 e-

"~ 10. r t#}

R 2 = 0.88 10

100

1000

B, Bed thickness (cm)

Fig. 16. An example of the linear relation between fracture spacing and bed thickness (from the Pennapiedimonte section).

fractures, whereas the Bolognano Fm. is characterised by both isolated and coupled discontinuities. In terms of fluid flow/storage properties, this suggests that fractures affecting the Tre Grotte Fm. may enhance fluid migration, whereas the Bolognano

Fm. can provide the appropriate conditions for their storage, because of the lack of sufficient connectivity. Model simulation. One of the main objectives of fault and fracture analysis is that of building up appropriate three-dimensional models of natural

128

L. MARCHEGIANI E T A L .

Fig. 18. Triangular diagram showing the connectivity properties of the Bolognano and Tre Grotte Formations. on fracture properties, as these have been determined by systematic analysis in outcrops. As a result of our work, a DFFN model of a layered rock volume of about 100 m 3 was set up with the aim of assessing the hydraulic properties of this potential geofluid reservoir hosted within the carbonate sequence of the Majella anticline, and to simulate the flow conditions during different production operations (Fig. 20). Fig. 17. Spatial distribution of P21 values. The plot was obtained by interpolating about 50 values of P21 derived from scan area analysis. As may be seen, high values of fracture density occur along the eastern limb of the anticline, whereas low values occur mainly at the northern periclinal termination.

discontinuity networks, in order to evaluate the fluid transport and storage properties of fractured rock volumes (Nurmi et al. 1973; Rawnsley & Wei 2001). In the last decades, two types of models have generally been used to this end: dual porosity models and DFFN models (Fig. 19). Dual porosity models simulate fractured rock volumes by means of two continuous domains, one made of matrix blocks, characterized by high fluid storage behaviour, and the other represented by interconnected fractures (i.e. empty spaces) that behave as conduits to fluid migration. These models are useful for predicting 'simple' relations for fluid flow behaviour; however, they do not take into account any real properties of natural fractures, such as their orientation, aperture, length, and so on. More accurate predictions can instead be obtained by using DFFN models, which accept as input data, statistical and probabilistic information

Fig. 19. Examples of (a) a dual porosity model and (b) a DFFN model.

FRACTURE SYSTEMS IN CENTRAL ITALY

129

may be described by a power-law relation with a negative exponent, equal to -1.88, over about four to five orders of magnitude. Other studies on natural faults exposed in different geological contexts show, however, that this value may be quite variable. For instance, Scholz et al. (1993) suggest a value of 1.3 for normal faults affecting the Bishop tuff in Owen's Valley, California; Scholz & Cowie (1990) and Tondi & Cello (2003) suggest a range of values between 1.1 and 1.5 for crustal-scale strike-slip and normal faults in Japan and in Italy respectively (see also van Dijk et al. 2000; Cello et al. 2003).

Fig. 20. DFFN model of a 100 m3 rock volume belonging to the stratigraphic sequence of the Cima delle Murelle Fm. The model was obtained by using a recently developed software that generates fault and fracture networks through an iteractive process. This is based on specific algorithms that 'fill in' the selected rock volume with structural discontinuities characterized by the power-law relations derived from field studies.

Summary and conclusions The Majella Mountain is a thrust-related fold dissected by a network of faults and fractures showing geometric and overprinting relationships with the anticlinal structure. These relations, as well as the main characteristics of the exposed fault and fracture networks, were sorted out and allowed us to evaluate the role of local geological and structural heterogeneities in controlling their systematic distribution and clustering within the different sectors of the Majella structure. The results of our study allowed us to assess the following points. (1)

(2)

(3)

The examined fracture populations belong to three main sets; their relative chronology indicates that the oldest set is the one trending roughly E - W , whereas younger sets are those oriented N W - S E and NE-SW. Fracture aperture clusters around a roughly NNW-SSE trend, which is parallel to the orientation of the present-day maximum horizontal stress component acting in the area; this stress state has been acting in peninsular Italy over the last 700,000 years. Outcrop observations also show that reactivation of the E - W and N W - S E trending structures occurred in response to this same stress field. Length distributions relative to both fractures and faults of different size indicate that they

Some of the observed variations of the value of the power-law exponent have been explained by invoking (1) a transition from small to large faults (see, for example, Cowie et al. 1993), (2) a bias due to sampling procedures (see Marret & Allmendinger 1991, among others), (3) a different degree of the finite state of deformation (see, for example, Sornette et al. 1993; Cowie et al. 1993). In particular, the latter authors show that, as a model material is increasingly deformed, the exponent of the power law decreases from a value of about 2.0 to 1.0. Accordingly, we suggest that the value of the exponent of the power law derived for the Majella area (-1.88) may be indicative of a low degree of maturity of the analysed faults and fractures. A similar result was obtained by Cello (1997), who analysed a quaternary fault system exposed in peninsular Italy, north of the Majella area. (4) Fracture spacing displays the following characteristics: (a) fractures in the protolith are quite regular (with Cv < 1) and closely spaced spacing is often correlated to bed thickness; (b) fault-related fractures show a general increase in frequency (about three times with respect to background values in the protolith) and a power-law distribution; (c) the spatial distribution of mesofaults is random, with a coefficient of variation Cv ~ 1. Similar results were obtained by Gillespie et al. (1993,2001), who found that fault-related fractures show a spacing distribution with a power-law exponent of between 0.4 and 1.0, whereas fractures in the protolith display regular spacing, and therefore cannot be considered to be fractals. (5) Fracture density was investigated at different scales by defining the spatial variations of P21 values within different sectors of the Majella anticline; our results show that the highest values occur along the flanks of the fold structure, whereas low values are characteristic of the fold hinge zones. This type of relation between fracture intensity and dip variations within different fold sectors was also emphasized by Hennings et aL (2000) and Johnson & Johnson (2000), who analysed an asymmetric anticline in the Casper arch (Wyoming) and a monoclinal fold

130

L. MARCHEGIANI ET AL.

in San Rafael swell (Utah), respectively. (6) The faults dissecting the Majella structure may be grouped into four types, according to their age, fault zone characteristics, and lithology. Our results show that, in general, fault displacement is systematically related to fault zone size, and its cumulative distribution is fractal with a power-law exponent of -0.85. We also noticed that the displacement distribution results were fractal despite the stratigraphic variations within faulted rock volumes, and the variable chronological and kinematic properties of the analysed faults. The conclusions arrived at for the Majella area, which, in our opinion, also apply to the majority of the Apennine structures and possibly to other carbonate settings worldwide, were then used to set up and test a three-dimensional discrete fault and fracture model. This allowed us to assess the fluid flow and storage characteristics of the selected rock volume, and to evaluate the associated fault seal risk and the vulnerability conditions of the reservoir. This work is a contribution to the Task Force Majella Project, and was co-funded by Eni-Agip, the University of Camerino, and by the MIUR, Cofin 2002 (research funds to G. C.). We wish to thank L. Mattioni and L. Micarelli for their help during fieldwork. The authors also wish to acknowledge the work done by former students A. Petritoli, B. Prugni, G. Zanni and G. Bellezza during the preparation of their graduate theses at the Department of Earth Sciences of Camerino University, and the staff of the Photogrammetric Group of Eni-Agip (TEIC) involved in the Majella Project.

References ANTONELLIm, M. & AYDIN, A. 1995. Effect of faulting on fluid flow in porous sandstones: geometry and spatial distribution. AAPG Bulletin, 79, 642-671. BAR, P. & TANG, C. 1989. Earthquakes as a self-organized critical phenomenon. Journal of Geophysical Research, 94, 15635-15637. BARTON, C. C. 1995. Fractal analysis of the scaling and spatial clustering of fractures in rock. In: BARTON, C. C. & LA POINTE, P. R. (eds) Fractals in the Earth Sciences. Plenum Press, New York 141-177. CAINE, J. S., EVANS, J. P. & FORSTER, C. B. 1996. Fault zone architecture and permeability structure. Geology, 11, 1025-1028. CELLO, G. 1997. Fractal analysis of a Quaternary fault array in the central Apennines, (Italy). Journal of Structural Geology, 19, 945-953. CELLO, G. 2000. A quantitative structural approach to the study of active fault zones in the Apennines (Peninsular Italy). Journal of Geodynamics, 29, 265 -292. CELLO, G., MAZZOLI, S., TONDI, E., & TURCO, E. 1997. Active tectonics in the central Apennines

and possible implications for seismic hazard analysis in peninsular Italy. Tectonophysics, 272, 43-68. CELLO, G., GAMBINI, R., MAZZOLI, S., READ, A., TONDI, E. & ZUCCONI, V. 2000. Fault zone characteristics and scaling properties of the Val d'Agri Fault System (Southern Apennines, Italy). Journal of Geodynamics, 29, 293- 307. CELLO, G., TONDI, E., MICARELLI, L. & INVERNIZZI, C. 2001. Fault zone fabrics and geofluid properties as indicators of rock deformation modes. Journal of Geodynamics, 32, 543-565. CELLO, G., TONDI, E., VAN DIJK, J. P., MATTIONI, L., MICARELLI, L. & PINTI, S. 2003. Geometry, kinematics and scaling properties of faults and fractures as tools for modelling geofluid reservoirs: examples from the Apennines, Italy. In: NmUWLAND, D. A. (ed.) New Insights into Structural Interpretation and Modelling. Geological Society, London, Special Publications, 212, 7-22. COWIE, P. A. 1998. A healing-reloading feedback control on the growth rate of seismogenic faults. Journal of Structural Geology, 20, 1075-1087. Cowm, P. A. & ROBERTS, G. 2001. Constraining slip rates and spacings for active normal faults. Journal of Structural Geology, 23, 1901 - 1925. COWIE, P. A., VANNESTE, C. & SORNETTE, D. 1993. Statistical physics model for the spatiotemporal evolution of faults. Journal of Geophysical Research, 98, 21809-21821. DI BtJccI, D. & MAZZOLI, S. 2002. Active tectonics of the Northern Apennines and Adria geodynamics: new data and a discussion. Journal of Geodynamics, 34, 687-707. DONZELLI, G. 1997. Studio geologico della Majella. PhD thesis, Universith degli Studi G. D'Annunzio, Dipartimento di Scienze della Terra. EBERLI, G. P., BERNOULLI, D., SANDERS, D. & VECSEI, A. 1993. From aggradation to progradation: The Majella platform (Abruzzi, Italy). In: SIMO, T., SCOTT, R. W. & MASSE, J. P. (eds) Atlas of Cretaceous Carbonate Platforms. American Association of Petroleum Geologists, Memoir 56, 213-232. EVANS, J. P., FOSTER, C. B. & GODDARD, J. V. 1997. Permeability of fault-related rocks and implication of hydraulic structure of fault zones. Journal of Structural Geology, 19, 1393-1404. FLETCHER, R. C. & POLLARD, D. D. 1981. Anticrack model for pressure solution surfaces. Geology, 9, 419-424. GHISETTL F. & VEZZANI, L. 2002. Normal faulting, extension and uplift in the outer thrust belt of the central Apennines (Italy): role of the Caramanico fault. Basin Research, 14, 225-236. GILLESPIE, P. A., WALSH, J. J. & WATTERSON, J. 1992. Limitations of dimension and displacement data from single faults and the consequences for data analysis and interpretation. Journal of Structural Geology, 14, 1157-1172. GILLESPIE, P. A., HOWARD, C. B., WALSH, J. J. & WATTERSON, J. 1993. Measurement and characterization of spatial distribution of fractures. Tectonophysics, 226, 113-141.

FRACTURE SYSTEMS IN CENTRAL ITALY GILLESPIE, P. A., WALSH, J. J., WATTERSON, J., BONSON, C. G. & MANZOCCHI, T. 2001. Scaling relationships of joint and vein arrays from The Burren, Co. Clare, Ireland. Journal of Structural Geology, 23, 183-201. GOES, S., GIARDINI, D., JENNY, S. HOLLENSTEIN, C., KAHLE, H.-G. & GEIGER, A. 2004. A recent tectonic reorganization in the south-central Mediterranean. Earth and Planetary Science Letters, 226, 335-345. GRAHAM, B., ANTONELLINI, M. & AYDIN, A. 2003. Formation and growth of normal faults in carbonates within a compressive environment. Geological Society of America, 31, 11 - 14. HENNINGS, P. H., OLSON, J. E. & THOMPSON, L. B. 2000. Combining outcrop data and three-dimensional structural models to characterize fractured reservoirs: an example from Wyoming. AAPG Bulletin, 84, 830-849. JOHNSON, K. M. & JOHNSON, A. M. 2000. Localization of layer-parallel faults in San Rafael swell, Utah and other monoclinal folds. Journal of Structural Geology, 22, 1455-1468. LAMPERT, S. A., LOWRIE, W., HIRT, A. M., BERNOULLI, D. & MUTTI, M. 1997. Magnetic and sequence stratigraphy of redeposited Upper Cretaceous limestones in the Montagna della Maiella, Abruzzi, Italy. Earth and Planetary Science Letters, 150, 79-83. LOOSVELD, R. J. H. ~r FRANSSEN, R. C. M. W. 1992. Extensional vs. shear fractures: implications for reservoir characterisation. European Petroleum Conference, 16-18 November. Proceedings 2, Society of Petroleum Engineers, Paper 25017. MARCHEGIANI,L., BERTOTTI, G., CELLO, G., DEIANA, G., MAZZOLI, S. & TONDI, E. 1999. Pre-orogenic tectonics in the Umbria-Marche sector of the afro-adriatic continental margin. Tectonophysics, 315, 123-143. MARIUCCI, M. T. &: Mt3LLER, B. 2003. The tectonic regime in Italy inferred from borehole breakout data. Tectonophysics, 361, 21-35. MARRET, R. t% ALLMENDINGER, R. W. 1991. Estimates of strain due to brittle faulting - sampling of fault populations. Journal of Structural Geology, 13, 735-738. MOLLER-PEDERSEN, P. & KOESTLER, A. G. (eds) 1997. Hydrocarbon Seals Importance for Exploration and Production. NPF, Special Publication, 7, Elsevier Science Ltd. NURMI, R. D., STANDEN,E. J. W. & AKBAR, M. 1993. Geometrical characteristics and geological modelling of fractured Middle East reservoirs. Society of Petroleum Engineers, Paper 25639.

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ORTEGA, O. 8Z MARRET, R. 2000. Prediction of macrofracture properties using microfracture information, Mesaverde Group sandstones, San Juan basin, New Mexico. Journal of Structural Geology, 22, 571588. RAWNSLEY, K. & WEI, L. 2001. Evaluation of a new method to build geological models of fractured reservoirs calibrated to production data. Petroleum Geoscience, 7, 23-33. SCHLISCHE, R. W., YOUNG, S. S., ACKERMANN,R. V. & GUPTA, A. 1996. Geometry and scaling relations of a population of very small rift-related normal faults. Geology, 24, 683-686. SCHOLZ, C. H. 8z COWIE, P. 1990. Determination of total strain from faulting using slip measurements. Nature, 346, 837-839. SCHOLZ, C. H. & GUPTA, A. 2000. Fault interactions and seismic hazard. Journal of Geodynamics, 29, 459-467. SCHOLZ, C. H., DAWERS, N. H. YU, J. Z. ANDERS, M. H. & COWIE, P. A. 1993. Fault growth and fault scaling laws - preliminary results. Journal Geophysics Research, 98, 21951 - 21961. SCISCIANI,V., TAVARNELLI,E. t~r CALAMITA,F. 2002. The interaction of extensional and contractional deformation in the outer zones of the central Apennines, Italy. Journal of Structural Geology, 24, 1647-1658. SIBSON, R. H. 2000. Fluid involvement in normal faulting. Journal of Geodynamics, 29, 469-499. SORNETTE, A., DAVY, P. • SORNETTE, D. 1993. Fault growth in brittle-ductile experiments and the mechanics of continental collisions. Journal of Geophysical Research, 98, 12111 - 12139. TONDI, E. & CELLO, G. 2003. Spatiotemporal evolution of the Central Apennines fault system (Italy). Journal of Geodynamics, 36, 113-128. VAN DIJK, J. P., BELLO, M., TOSCANO, C., BERSANI, A. 8z NARDON, S. 2000. Tectonic model and three-dimensional analysis of Monte Alpi (southern Apennines). Tectonophysics, 324, 203-237. WALSH, J. J. 8z WATTERSON, J. 1988. Analysis of the relationship between displacements and dimensions of faults. Journal of Structural Geology, 10, 239-247. WALSH, J. J. & WATTERSON, J. 1993. Fractal analysis of fracture patterns using the standard boxcounting technique: valid and invalid methodologies. Journal of Structural Geology, 15, 1509-1512. YIELDING, G., NEEDHAM, T. & JONES, H. 1996. Sampling of fault populations using sub-surface data: a review. Journal of Structural Geology, 18, 135-146.

Evidence for the existence of a simple relation between earthquake magnitude and the fractal dimension of seismogenic faults: a case study from central Italy G. CELLO, L. M A R C H E G I A N I & E. T O N D I

D i p a r t i m e n t o di Scienze della Terra, University o f Camerino, Via Gentile III da Varano, 62032 C a m e r i n o (MC), Italy (e-mail: giuseppe, cello @ unicam.it)

Abstract: Fault data from the central Apennines (Italy) were integrated with earthquake information from seismic catalogues in order to derive an empirical relation between the magnitude of the strongest historical earthquake and the fractal dimension of active fault patterns. We show that the assessment of earthquake magnitude from fault data has given good results, hence suggesting that the relation may be used to evaluate the potential hazard of seismic source areas in the Apennines using a low-cost methodology. We also suggest that a similar approach may be used in other seismic belts worldwide, provided that the basic seismological and geological information needed is adequate to constrain the appropriate relation between these two size parameters.

Our current knowledge of earthquake faulting (e.g. Cello & Tondi 2000; Scholz 2002) suggests that the assessment of the scaling properties of fault zones by means of fractal statistics, and the simulation of their growth by means of Self Organized Criticality (SOC) models (e.g. Cowie et al. 1993; Bak & Tang 1995) provide interesting perspectives for seismic hazard evaluation (e.g. Main 1995; Sherman & Gladkov 1999; Cowie & Roberts 2001; Tondi & Cello 2003). In the last couple of decades, systematic studies of fault zone characteristics have significantly improved our understanding of the factors controlling their spatial arrangement and geometric complexity (e.g. Caine et al. 1996; Cello et al. 2000a, b; Aydin et al. 2005). At the same time, SOC models have emphasized that fault growth occurs, over long time-scales, through the coalescence of distributed lower-rank structural features that eventually link together and localize strain on a few dominant fractal structures that control most of the associated seismic energy release (e.g. Sornette & Sornette 1989; Sornette et al. 1990a; Cowie et al. 1993). Furthermore, the transition from distributed to localized deformation within a brittle shear zone seems to be marked by its tendency to change from an Euclidean to a fractal geometry (Sornette et al. 1990b; Cello 1997). As a result, the degree of complexity of a finite fault zone (which can be quantified by measuring its fractal dimension, D) is considered to be an indicator of fault size at any given evolutionary stage in the space-time domain. Accordingly, we suggest that measuring the spatial pattern of active faults within a seismic belt may be used to predict

earthquake size, as it is well established that fault dimension controls the seismic moment release, and hence earthquake magnitude.

Fault and earthquake data The central sectors of peninsular Italy are part of the Apennines fold and thrust belt, and include a few major tectonic elements of the peri-Mediterranean mountain system (Fig. 1). These sectors of the Apennines, which formed in response to the convergent motion of the African and European plates (Dewey et al. 1989; Goes et al. 2004), are made up of different tectonic units derived from the deformation of the various palaeogeographic domains of the southern sectors of the Afro-Adriatic continental margin (see e.g. Calamita et al. 1994a; Deiana & Pialli 1994). In central Italy, a Plio-Quatemary fault system dissects older structural fabrics of the former (Late Miocene-Early Pliocene) fold and thrust belt (Fig. 2, see also Cello et al. 1997; Barchi et al. 2000; Boncio et al. 2000). The faults belonging to this system, showing different kinematic behaviour and variable ages, include newly generated and reactivated features of the TyrrhenianApennines domain that formed during previous evolutionary stages in the long deformation history of the area (see Calamita et al. 1994b; Cello et al. 1997). Field work and remote sensing analysis of the major faults exposed in the axial zones of the central Apennines allowed us to recognize several active segments and to derive a detailed fault

From: CELLO,G. & MALAMUD,B. D. (eds) 2006. Fractal Analysis for Natural Hazards. Geological Society, London, Special Publications, 261, 133-140. 0305-8719/06/$15.00 9 The Geological Society of London 2006.

134

G. CELLO E T A L . I 13~

Adriatic

Sea

Ancona Roma

1t

Bari Tyrrhenian Sea

N

civitaw 0

45 km

Tuscan nappe Cervarola unit Gran Sasso unit

Rom,

Umbria-Marche units Periadriaic foothill units ~*-C] Volcanics

Fig. 1. Geo-tectonic setting of the study area, in Italy.

map of the area. This includes seven kinematically coherent arrays, which were interpreted as the surface expression of the well-known seismogenic structures thought to be responsible for most of the earthquakes occurring in this sector of the mountain belt (Fig. 3). These arrays, as a whole, make up the so-called Central Apennines Fault System (CAFS) and are the causative structures generating earthquakes with magnitudes in the range 5.0-7.0 (Table 1). They also control the evolving fragmentation pattern of the brittle crustal volume undergoing deformation in response to the current stress regime acting in the area from about 700,000 years ago (Cello et al. 1997; Di Bucci & Mazzoli 2002; Goes et al. 2004). The record of the major historical earthquakes that struck this sector of the Apennines includes mostly shallow ( < 2 0 k m deep) and a few intermediate (< 100 km deep) events (Amato & Selvaggi 1991). Destructive earthquakes in the area are mainly confined within intramontane Plio-Pleistocene basin areas located within the CAFS, and most of them are generated at shallow depth in the crust. The scaling relation between computed seismic moments and length of the inferred CAFS-related seismogenic faults (Fig. 4) shows that most of the events may be considered as 'small' earthquakes (Scholz 2002) following the expression M0 ~ L 3

(with M0 = seismic moment and L = fault length). This result suggests therefore that the analysed earthquakes are characterized by source dimensions that are smaller than the inferred thickness of the seismogenic layer (about 12kin; Deschamps et al. 1984). The log N v. M relation (with N = number of earthquake and M = earthquake magnitude) is characterized by a b value of 0.8 (Fig. 5). This suggests that the magnitude of the maximum expected event, in the area, is of the same order as the largest historical event recorded in central Italy in the last millennium. In order to derive an empirical relation between earthquake magnitude and the D-value of fault map patterns, we analysed in detail the geological database available for the CAFS (see Tondi & Cello 2003) and the most recent seismic catalogues compiled for this sector of the Italian Peninsula (Boschi et al. 1997; Camassi & Stucchi 1998; C.P.T.I. Gruppo di Lavoro 1999). This allowed us to identify, for each of the seven seismogenic zones of Figure 3, the strongest historical earthquake and to assess its equivalent magnitude (Me). The fractal dimension (D) of the different fault arrays (each corresponding to a single earthquake source area) was obtained by using the

EARTHQUAKES AND FRACTAL DIMENSION

Fig. 2. Structural features of the Central Apennines Fault System: (a) Simplified fault map and seismic activity (1985-2000) in the study area; (b) representative geological profile and fault structure across the Colfiorito seismogenic zone.

135

136

G. CELLO E T A L .

Fig. 3. Map of active faults in the axial zones of the central Apennines, Italy.

EARTHQUAKES AND FRACTAL DIMENSION

Table 1. Historical earthquakes from Boschi et al.

1.4

1997: Characteristic parameters of Central Apennines Fault System-related earthquakes and associated seismogenic zones

1.2

137

1.0 0.8

Year

Epicentral area

/max

Me

Seismogenic fault zone

1599 1639 1703 1730 1859 1979 1915 1997

Cascia Amatrice Norcia Leonessa Vettore Norcia Fucino Colfiorito

8.0 9.0 11.0 7.5 7.0 8.5 11.0 8.0

5.5 5.4 6.7 5.0 5.0 5.9 6.9 5.9

4 6 3 5 2 3 7 1

~0.6 0.4 0.2

0.0 5.0

10~6 E ._ mE10~

5.5

\

6.0

6.5

\ 7.0

7.5

Fig. 5. Cumulative distribution of the equivalent magnitudes (Me) of the historical earthquakes occurring within the Central Appennines Fault System (data from Tondi & Cello 2003).

10~7 = 5xt022 x L25

".C

(N) = -0.8 Me + 5.6~

Me

box-counting technique, a conventional method for analysing map patterns, that is, the two-dimensional spatial properties of a given structure (Mandelbrot 1983). This technique allows one to derive log/ log diagrams with Ns (number of boxes containing the pattern) plotted as a function of s (where s is the size of the measuring grid) and to construct boxcounting curves whose slope values give the appropriate fractal dimension of the analysed structure. The D values obtained from the box-counting curves derived from the map of the seven seismogenic fault zones of the CAFS are shown in Figure 6. In Figure 7, we plotted the D. value together with the appropriate Me values inferred for each earthquake source area. As may be seen, except for point 3* in Figure 7, the two size parameters display a well-constrained relation. This result is relevant, in our opinion, for the evaluation of earthquake-related hazard, as the relation Me = l l D 7 may lead to accurate predictions of the seismogenic potential of a given fault zone. The fact that point 3* does not fit the correlation curve and plots below it is also of interest, as it suggests that the 1979 Norcia earthquake (Me = 5.9), which occurred within the seismogenic

Mo

i•Log I

zone 3, did not release all the strain energy available in the system. This is in agreement with available historical information on the seismic activity in the Norcia zone, as there is a lot of evidence that the whole area was struck by a Me > 6.5 earthquake in 1703 (Boschi et al. 1997; Cello et al. 1998). As shown in Figure 7, this latter event (point 3) plots much closer to the M e - D correlation curve than point 3*. Therefore, our study not only shows that there exists a linear relation between Me and D, but also that 'anomalous' values (i.e. point 3* in Fig. 7) indicate that each seismogenic zone may release variable amounts of stored strain energy through the occurrence of earthquakes with Me values smaller than that of the maximum expected event. This may be better appreciated by comparing the earthquake potential of the seismogenic zones (3: Norcia) and (1: Colfiorito). In these cases, it may be observed that although the 1979 earthquake in the Norcia area is a lowenergy event (Me = 5.9) within an active fault zone with a seismic potential of Me > 6.5, the 1997 Colfiorito earthquake (Me = 6) may be considered as the strongest event that may possibly be generated within the seismogenic zone 1.

Conclusions

f

t

10 Fault length (L)

Fig. 4. Fault length v. seismic moment of the Central Apennines Fault System-related historical earthquakes (data from Tondi & Cello 2003).

100

The results of this study emphasize that our assumption suggesting that it is possible to obtain information on the seismogenic potential of a given area by measuring the fractal dimension of active faults is fundamentally correct. In our opinion, this is not surprising because (1) the fractal dimension of fault map patterns represents a measure of fault complexity and size, and hence of the degree of maturity of a fault zone (Cello 1997); (2) it is well established that fault growth processes occur mainly through linkage between fault segments (Cartwright et al. 1995; Cladouhos & Marrett 1996). Consequently, any

138

G. CELLO

ETAL.

Zone 1 10000.

10o0

Zone 2

3

N(s) = 2108s 1"23 1000 lOO

I1\ 1~176

N(s) 10

1~1 1

,

I

10

100

1000

10

100 S Zone 4

S

Zone 3

10000

I0000

N(s) = 1000

1000

N(s) lO0

N(s9 100

1

10

10000

S Zone 5

100

100

S

10000

Zone 6

N(s) = 1279s1"- 1 6

N(s) : 1241s 1"13 1000

1000

N(s) 100

N(s) t00

lO

139 ls 1" 18

i0

1000

1000

lOO

1000

S

10

100

1000

S

Zone 7

10000'

~

N

(

s = 7339s ) -1"30

Zone 1: Colfiorito Zone 2: Mt. Vettore

1000

Zone 3: Norcia

N(s) 100

Zone 4: Cascia Zone 5: Leonessa Zone 6: Mt. Gorzano Zone 7: Fucino 10

s

100

1000

Fig. 6. Box-counting curves (the number of boxes N at a given box size s are plotted as a function of s, on log-log axes) of the analysed active fault patterns shown in Figure 3.

EARTHQUAKES AND FRACTAL DIMENSION

139

References

7.0

6.5-

.....

AMATO,A. & SELVAGGI,G. 1991. Terremoti crostali e

iii:'i..';

i

6.0,.* " 5,5-

5.0-

.... 6

~,t' %* 4

1

3*

. . . . . .

.....

Me =

5 2 i

45

11D-7

......

! 1.10

1.15

1.20

1.25

1.30

1.35

D

7. Diagram showing the relation between the equivalent magnitude (Me) of historical earthquakes occurring in the axial zones of the central Apennines and the fractal dimension (D) of the seismogenic faults shown in Figure 3. Fig.

increment in fault dimensional properties and geometric complexity produces an increase in the D value of each fault pattern; (3) any increment in size of an actively faulting rock volume (i.e. of a seismogenic source area) causes an increase of the expected maximum earthquake moment, and hence of its equivalent magnitude (Me). We are aware that in different geostructural contexts one may, however, not be able to collect all the necessary geological and/or seismological information needed to derive an appropriate M e - D relation. Possible limitations to a generalized use of the proposed procedure for assessing the earthquake hazard of a given area may in fact come from (1) poor resolution in the identification of seismogenic sources within a seismic belt, (2) scarcity of good historical and/or instrumental records of earthquakes, and (3) our ability to recognize and map active fault segments with the appropriate details required for standard box-counting analysis. In conclusion, we have shown that predicting earthquake magnitude from fault data has given good results for the axial zones of the central Apennines; we believe, however, that more data from other areas of active faulting worldwide are needed to possibly generalize the empirical relation proposed in this study.

The authors are grateful to B. Malamud, S. Pavlides and L. Telesca for helpful reviews. This work has been supported by the University of Camerino (research funds to G. C. and E. T.) and by the MIUR, Cofin 2002 (research funds to G. C.).

sub-crostali nell'appennino settentrionale. Studi Geologici Camerti, 1, 75-82 (Volume Speciale). AYDIN, A., BORJA, R. I. & EICHHUBL, P. 2005. Geological and mathematical framework for failure modes in granular rocks. Journal of Structural Geology, 28, 83-98. BARCHI, M., GALADINI, F. ET At. 2000. Sintesi delle conoscenze sulle faglie attive in Italia Centrale: parametrizzazione ai fini della caratterizzazione della pericolosith sismica. CNR-Gruppo Nazionale per la Difesa dai Terremoti - Roma. BAK, P. & TANG, C. 1989. Earthquakes as a self-organized critical phenomenon. Journal of Geophysical Research, 94, 15635-15637. BONCIO, P., BROZZETTI, F. & LAVECCHIA,G. 2000. Architecture and seismotectonics of a regional low-angle normal fault zone in central Italy. Tectonics, 19, 1038-1055. BOSCHI, E., GUIDOBONI,E., EERRARI, G., VALENSISE, G. & GASPERINI,P. 1997. Catalogo deiforti terremoti in Italia dal 461 a.C. al 1990, 2, Instituto Nazionale di Geofisica and Storica Geofisica Ambiente, Bologna, 644 p. and CD ROM. CAINE, J. S., EVANS, J. P. & FORSTER, C. B. 1996. Fault zone architecture and permeability structure. Geology, 11, 1025-1028. CALAMITA, F., CELLO, G., DEIANA, G. & PALTRIN1ERI, W. 1994a. Structural styles, chronology rates of deformation, and time-space relationships in the Umbria-Marche thrust system (central Apennines, Italy). Tectonics, 13, 873-881. CALAMITA, F., COLTORTI, M., FARABOLLINI, P. & lhzz~, A. 1994b. Le faglie normali quaternarie nella Dorsale appenninica umbro-marchigiana: proposta di un modello di tettonica d'inversione. In: LAZZAROTTO,A. & LIOTTA, D. (eds) Studi Preliminiaric all'Acquisizione Dati del Profilo CROP 18 Lardarello-Mite Amiata. Studi Geologici Camerti, Volume Speciale, 211-225. CAMASSI, R. & STUCCHI, M. 1998. NT4.1, a parametric catalogue of damaging earthquakes in the Italian area (release NT4.1.1). Gruppo Nazionale per la Difesa dai Terremoti, Milano. World Wide Web Address: http://emidius.mi.ingv.it/NT/ frontespizio.html [Accessed 1 July 2006]. CARTWRIGHT, J. A., TRUDGILL, B. D. & MANSFIELD, C. S. 1995. Fault growth by segment linkage an explanation for scatter in maximum displacement and trace length data from the Canyonlands Grabens of SE Utah. Journal of Structural Geology, 17, 1319-1326. CELLO, G. 1997. Fractal analysis of a Quaternary fault array in the central Apennines, Italy. Journal of Structural Geology, 19, 945-953. CELLO, G. & TONDI, E. (eds) 2000. The resolution of geological analysis and models for earthquake faulting studies. Journal of Geodynamics, 29, 3-5. CELLO, G., MAZZOLI, S., TONDI, E. & TURCO, E. 1997. Active tectonics in the central Apennines and possible implications for seismic hazard analysis in Peninsular Italy. Tectonophysics, 272, 43-68.

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CELLO, G., MAZZOLI, S. & TONDI, E. 1998. The crustal fault structure responsible for the 1703 earthquake sequence of central Italy. Journal of Geodynamics, 26, 443-460. CELLO, G., GAMBINI, R., MAZZOLI, S., READ, A., TONDI, E. & ZUCCONI, V. 2000a. Fault zone characteristics and scaling properties of the Val d'Agri Fault System (Southern Apennines, Italy). In: CELLO, G. & TONDI, E. (eds) The Resolution of Geological Analysis for Earthquake Faulting Studies. Journal of Geodynamics, 29, 293-307. CELLO, G., DEIANA, G. ET AL. 2000b. Geological constraints for earthquake faulting analysis in the Colfiorito area. Journal of Seismology, 4, 357-364. CLADOUHOS, T. T. & MARRETT, R. 1996. Are fault growth and linkage models consistent with power-law distributions of fault lengths? Journal of Structural Geology, 18, 281 - 293. Cowm, P. A., VANNESTE, C. & SORNETTE, D. 1993. Statistical physics model for the spatiotemporal evolution of faults. Journal of Geophysical Research, 98, 21809-21821. Cowm, P. A. & ROBERTS, G. 2001. Constraining slip rates and spacings for active normal faults. Journal of Structural Geology, 23, 1901-1925. C.P.T.I. GRUPPO D] LAVORO 1999. Catalogo Parametrico dei Terremoti Italiani. ING, GNDT, SGA, SSN, Bologna, 1999. DEIANA, G. & PIALLI, G. 1994. The structural provinces of the Umbro-Marchean Apennines. Memorie della Societgt Geologica Italiana, 48, 473-484. DESCHAMPS, A., IANNACCIONE, G. & SCARPA, R. 1984. The Umbrian earthquake (Italy) of 19 September 1979. Annales Geophysicae, 2, 29-36. DEWEY, J. F., HELMAN, M. L., TURCO, E., HUTTON, D. H. W. & KNOTT, S. D. 1989. Kinematics of the Western Mediterranean. In: COWARD,

M. P., DIETRICH, D. & PARK, R. G. (eds) Alpine Tectonics, Special Publication, 45, 265-283. DI BuccI, D. & MAZZOLI, S. 2002. Active tectonics of the Northern Apennines and Adria geodynamics: new data and a discussion. Journal of Getdynamics, 34, 687-707. GOES, S., GIARDINI, D., JENNY, S., HOLLENSTEIN,C., KAHLE, H.-G. & GEIGER, A. 2004. A recent tectonic reorganization in the south-central Mediterranean. Earth and Planetary Science Letters, 226, 335- 345. MAIN, I. G. 1995. Seismogenesis and seismic hazard. In: RANALLI, G. (ed.) VIII Summer School Earth and Planetary Sciences on 'Plate Tectonics: The First Twenty-Five Years'. University of Siena Press, Siena, 395-419. MANDELBROT, B. B. 1983. The Fractal Geometry of Nature. Freeman, New York. SCHOLZ, C. H. 2002. The Mechanics of Earthquakes and Faulting, 2nd edn. Cambridge University Press, Cambridge. SHERMAN, S. I. & GLADKOV, A. S. 1999. Fractals in studies of faulting and seismicity in the Baikal rift zone. Tectonophysics, 308, 133-142. SORNETTE, A. & SORNETTE, D. 1989. Self-organized critically and earthquakes. Europhysics Letters, 9, 197-202. SORNETTE, D., DAVY, P. & SORNETTE, A. 1990a. Structuration of the lithosphere as a self-organised critical phenomenon. Journal of Geophysical Research, 95, 17353-17361. SORNETTE, A., DAVY, P. & SORNETTE, D. 1990b. Growth of fractal fault patterns. Physical Review Letters, 65, 2266-2269. TONDI, E. & CELLO, G. 2003. Spatiotemporal evolution of the central apennines fault system (Italy). Journal of Geodynamics, 36, 113-128.

Power-law extreme flood frequency R. K I D S O N 1., K. S. R I C H A R D S 2 & P. A. C A R L I N G 3

1Trinity College, Cambridge CB2 1TQ, UK* (e-mail: [email protected]) 2Department of Geography, University of Cambridge, Downing Place, Cambridge CB2 3EN, UK (e-mail: [email protected]) 3Department of Geography, Highfield, Southampton University, S 0 1 7 1B J, UK (e-mail: P.A. Carling@ soton.ac, uk) *Present address: N S W Department of Natural Resources, PO Box 651, Level 1 High Street Penrith, N S W 2751, Australia (e-mail: R.L.Kidson.O1 @Cantab.net) Abstract: Conventional Flood Frequency Analysis (FFA) has been criticized both for its questionable theoretical basis, and for its failure in extreme event prediction. An important research issue for FFA is the exploration of models that have theoretical/explanatory value as the first step towards more accurate predictive attempts. Self-similar approaches offer one such alternative, with a plausible theoretical basis in complexity theory that has demonstrable wide applicability across the geophysical sciences. This paper explores a simple self-similar approach to the prediction of extreme floods. Fifty river gauging records from the USA exhibiting an outlier event were studied. Fitting a simple power law (PL) relation to events with return period of 10 years or greater resulted in more accurate discharge and return period estimates for outlier events relative to the Log-Pearson III model. Similar success in predicting record events is reported for 12 long-term rainfall records from the UK. This empirical success is interpreted as evidence that self-similarity may well represent the underlying physical processes generating hydrological variables. These findings have important consequences for the prediction of extreme flood events; the PL model produces return period estimates that are far more conservative than conventional distributions.

Introduction

A difficulty..,is the occurrenceon someriversof verylargefloods whichlie aboveand outsideof the uppercontrolcurve. (Gumbel 1958; p. 157) The main purpose of conventional flood frequency analysis (FFA), and certainly its key utility to society, is the prediction of the frequency and magnitude of extreme events in a given catchment. However, conventional analyses perform less than admirably at this task. This is particularly the case when there are multiple generating mechanisms (rainfall runoff and snowmelt, e.g. Waylen 1985), epochal variations in rainfall and runoff (Kiem et al. 2003), and outlier events, possibly associated with these phenomena, for which true return periods are poorly constrained. Conventional analyses perform poorly in these circumstances, and often predict return periods for extreme outlier events that are highly unrealistic and have led various authors (e.g. Baker 1994; Bardsley 1994) to be critical of conventional FFA. Klemes (1989) termed these the 'improbable probabilities' of extreme events. A reasonable conclusion from this evidence, which is further reinforced in this paper, is that the probability

model used has little predictive value for extreme events in many cases. Considering that large floods are some of the most common and destructive of natural hazards on the planet, and given the raison d'etre of FFA, this lack of predictive skill is unfortunate. It is perhaps not surprising, however, given that with the exception of Chow's (1954) theory of 'multiplicative processes' giving rise to hydrological variables that conform to a log-normal distribution, the majority of FFA distributions in present use articulate no basis in hydrological theory as justification for their application (Singh & Strupczewski 2002). An inspection of the literature confirms that the utility of the plethora of distributions available is judged largely on the basis of empirical goodness of fit (Alila & Mtiraoui 2002). Conventional FFA is largely a sophisticated exercise in parameter calibration, possessing neither explanatory nor, it appears, predictive power. It is reasonable to conclude, therefore, that the key research need in the field of FFA is in the investigation of models that promise some theoretical/explanatory basis (Hall & Anderson 2002) as the first prerequisite for real predictive value (Sivapalan 2003).

From: CELLO,G. & MALAMUD,B. D. (eds) 2006. FractalAnalysisfor Natural Hazards. Geological Society, London, Special Publications, 261, 141-153. 0305-8719/06/$15.00 9 The Geological Society of London 2006.

142

R. KIDSON ETAL.

This paper presents an alternative method for extreme event prediction, based on a self-similar approach. Data were collected for 50 catchments in the USA for which the records of annual maximum floods (record lengths ranging from 51 to 107 years) contain an outlier event. The officially mandated FFA distribution in the USA is the LogPearson III (LP3) (IACWD 1982). For each of these records, the LP3 distribution was fitted to all of the data with the exception of the outlier event; the model so fitted was then used to calculate the predicted return period of the observed outlier event (see Methods). This curve-fitting was then compared with the results of fitting a simple power-law relationship. The self-similar behaviour, which has been observed to be an emergent property of complex systems (Gupta 2004), is manifested as a straight line in double-logarithmic plotting space - a power-law (PL) relation. Applied to a magnitude/frequency curve of annual flood events, a PL relation takes the form

Q(T) = C T '~

(1)

where Q(T) is the discharge function of return period T and constants C and oz are conveniently found by regression in log-log space. Aside from this apparent analytical simplicity, self-similar processes possess a plausible theoretical basis in complexity theory and fractal and multifractal statistics (Bak 1996; Feder 1988; Mandelbrot & Wallis 1969), which describe the frequency distribution of events constituted from the superimposition of processes operating at a range of scales (Kirkby 1987; Koutsoyiannis 2003). These approaches have found successful predictive and descriptive application across a wide range of natural phenomenon. Turcotte (1994) presents case studies from a variety of natural phenomena including earthquakes and volcanic eruptions; Scheidegger (1997) provides examples from landslides, Birkeland & Landry (2002) from avalanches, Tzanis & Makropoulos (2002) from earthquakes, and Malamud & Turcotte (1999) from forest fires. Recently, self-similar approaches have also been applied in geomorphology (e.g. topography, Chase 1992) and hydrology, both for the case of rainfall events (Deidda et al. 1999; Ferraris et al. 2001) and flood events (Malamud & Turcotte 2006). Although this indicates recent growth of empirical evidence of self-similar environmental phenomena, there remains uncertainty and controversy about theoretical interpretations, and the following section briefly illustrates this, as well as indicating a theoretical rationale for the details of the methodology adopted for the analysis of flood series in this paper.

Emergent simple scaling One interpretation of complexity theory is that selfsimilar behaviour in nature can often be manifested not just in a single unique PL relation (the monofractal case), but in a multifractal scenario, where a specific PL relation applies only over a specific scale range (for a detailed mathematical description see Rodriguez-Iturbe & Rinaldo 1997). The single PL relation is a special case of the generalized PL form of the multifractal magnitude/frequency distribution: P

~

1~- k ( y )

(2)

(Lovejoy & Schertzer 1995), where P = exceedance probability, h is the variable of interest (e.g. peak annual river discharge), and k(y) is the exponent of the PL relation, which is itself a function. This exponent function can be visualized as either varying abruptly with discharge (producing a segmented PL plot), or with generating process. In the latter case, where there are several generating processes operating over the same discharge scale, the observed exponent will in fact be a composite. However, beyond some discharge value, one process may dominate, in which case the observed exponent is a real measure of the scale dependency of that process. Although providing an appropriate mathematical framework, this presents the analyst of a data-poor phenomenon with a confounding array of possible PL exponents over different scale ranges. It is well understood that floods at any given locality can be generated by different processes (Waylen & Woo 1982; Waylen 1985), each of which may conceivably possess its own fractal dimension; rarely are the distinct processes that generate extreme floods understood a priori. However, if the proposed scale range of interest is tightly defined, a local approximation (Sivakumar 2004) may be made, which assumes statistical simple scaling (Gupta 2004) - in short, a single PL relation. In making this assumption in relation to annual floods, the following qualifiers are appropriate. First, many of the potentially fractal physiographic factors contributing to a catchment's multifractal flooding characteristics may be held to be constant and characteristic of that catchment, for example slope, soil type and channel drainage network. Real physical processes typically involve such 'mixing' of individual fractal processes operating over the same scale range (Schertzer et al. 1991) and representable with a single (but approximate) PL relation. Rodriguez-Iturbe & Rinaldo (1997) cite two geomorphological examples where a technically multifractal phenomenon (stream drainage networks and the distribution of topographic

POWER-LAW EXTREME FLOOD FREQUENCY slopes) could, for analytical purposes, be collapsed to a single PL relation. Secondly, the probability of a single PL relation holding is enhanced if a limited scale range of discharges is defined; for example, translated to return period, it might be plausible to predict the 100-year flood using this approach, but not the 1000-year flood. This is particularly likely because longer return periods introduce the potential for significant climatic change between realizations of such events. Although still a contentious idea, Schertzer et al. (1993) propose that an important generic property of a fractal or multifractal distribution is the phase transition, above which PL behaviour becomes manifest. If substantiated and accepted, this phenomenon has potentially significant consequences, because it suggests that at the lower end of the scale range, PL behaviour should not be expected, depending upon the magnitude of the phenomenon at which the phase transition threshold occurs. Above this threshold, PL behaviour represents a major departure from conventional FFA distributions, which are characterized by asymptotic tail behaviour (parallel to the 'return period' axis) in untransformed space, and concave-down shape in l o g - l o g space. This is illustrated by the example shown in Figure 1. The main consequence of the PL, rather than concave-down tail behaviour in a distribution, is the far more conservative estimate of return period of the former relative to the latter, a fact that has very significant consequences for the prediction of the magnitude of extreme floods.

T"I

/

/

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/ /

] 10 4

Fig. 1. Observedannual flood events for the Cheat River, West Virginia (USGS Gauge No. 03069500), for the years 1914-2002, plotted in log-log space. Three conventionaldistributions are fitted to the observed data: the Gumbel (fitted with the Maximum Likelihood (ML) method), the two-parameter Log-Normal (LN2) distribution (fitted with the method of moments (MOM)), and the Log Pearson Type III (LP3) distribution, fitted with MOM and the WRC skew coefficient.

143

The properties of self-similar distributions have been identified empirically by studying data-rich phenomena (e.g. channel networks, hiUslopes) where continuous data are available over a relatively large range of scales (Rodriguez-Iturbe & Rinaldo 1997). Unfortunately, flood records are not data rich (because of their generally short length). Therefore, while it might be hypothesized that extreme events represent the manifestation of self-similar flood behaviour above an unspecified threshold (the phase transition), with the nonextreme floods comprising the below-threshold nonself-similar data, this is difficult to demonstrate at any given site, due to the rarity of extreme events at most yielding only one or two such events per locality. However, there are several reasons to infer the plausibility of this hypothesis. The first is the established self-similar behaviour noted in various scales of rainfall data, a major precursor to floods. Hubert (2001) and Hubert et al. (2002) have found evidence both of a phase transition and a PL tail when long-term annual total rainfall records from Europe are plotted in magnitude/frequency diagrams. One phase transition occurs at a 16-day time-scale related to atmospheric structures of planetary spatial extent, and different scaling transfer functions between rainfall and runoff occur in the high- and low-frequency regimes. A second reason is that support is provided by a small number of precedent flood studies. The first of these is an FFA conducted by Malamud et al. (1996), who fitted a PL relation to peaks-overthreshold flood data (the partial duration series) from the Mississippi River, and not only closely retrodicted the observed return period of the extreme flood event of 1993, but also retrodicted several palaeofloods of large magnitude from the Colorado River using the same approach. Use of the partialduration series overcomes some of the curvature in l o g - l o g space at the low end of the range associated with using the annual series, which is spuriously introduced by the restriction of the return period scale to a minimum value of one year in the annual series. A second set of results was presented by Potter (1958) and Potter et al. (1968), who studied the magnitude/frequency curves of a large number of small US catchments. Potter (1958) consistently identified a distinct change in gradient at around the 10 year return period, above which the magnitude/frequency relation exhibited straight-line behaviour on Gumbel distribution probability paper (Fig. 2). Potter did not plot these data in double logarithmic space; however, these findings are consistent with both the concept of a phase transition threshold, and PL tail behaviour thereafter. There seems to be no a priori reason to suspect

144

R. KIDSON E T A L .

24 2220

Upperfrequencycu

18-

1614(~ 12QI~

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N

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8-

Lowerfrequencycurve 4- j

\

_~

_. Q10(L) 5.7 =

20 1.01

i

2

5

10

50

Recurrence interval (years)

Fig. 2. A phase transition threshold observable in the magnitude/frequency curve. Potter (1958) studied a large number of catchments in the USA and identified a distinct change in gradient of the curve around a return period of 10 years (reproduced from Potter et al. 1968).

that the phase transition threshold should hold at a similar return period value (e.g. around l0 years) for a large range of catchments; however, using this threshold in annual series data does overcome the curvature issue affecting the analysis by Malamud et al. (1996). Potter's identification of this threshold immediately suggests a method for testing the applicability of a PL model for the prediction of extreme flood events. For a given gauging record exhibiting an outlier event, a PL relation may be fitted to all events with an observed return period greater than or equal to 10 years (excluding the outlier event(s)), and the success of the PL model in predicting the outlier event may then be calculated. This paper applies this simple approach to the 50 rivers listed in Table 1, using the methods discussed in the next section.

Methodology Peak annual instantaneous discharges (the annual maximum series) for the 50 US rivers identified in Table 1 were obtained from USGS (2005). The following criteria were applied to river selection: (1)

the longest available gauged records were sought; (2) only unregulated catchments were selected; (3) records exhibiting a flood caused by ice jamming and dam outbursts were not included, as it was desired to test the hypothesis for meteorologically derived flood events. Only records that exhibited an outlier event were included; although there are various methods for detecting outliers (Hawkins 1980), these methods have in common the fact that the definition of 'outlier' must be made relative to an a p r i o r i decision about what distribution(s) the events are sourced from. Here, we define an outlier event as one of a magnitude exceeding four standard deviations from the mean, and list the details of each record in Table 2. The average standard deviation (s.d.) displacement from the mean was 6.25, with a range of 4.74-8.99. This approach did not exclude records with two outliers; in 28 cases, the ratio of the outlier event to the nextranked one was less than two. The events identified are among the most extreme recorded in the history of instrumented river gauging in the USA. Data were obtained from most hydrologic regions, in order to provide as wide a geographical coverage as possible. No restriction was placed on basin size. The Mississippi River record studied by Malamud et al. (1996) did not qualify for inclusion in this data set on the > 4 s.d. criterion; however, results were also calculated for this record for the sake of comparability. The annual flood maxima for each river were ranked (including the outlier event), and observed return periods were assigned using the Weibull plotting position (i.e. i / ( n + 1)) where i = rank. This plotting position formula was selected in order to provide a simple unbiased exceedance probability independent of any distribution (Stedinger et al. 1993). The plotting position of the top-ranked event is sensitive to choice of plotting formula; these formulae are designed to correct bias, that is, systematic underestimation of the return period of the top-ranked event. This factor is recognized and dealt with in an alternative (probabilistic) fashion in the Results section. Both the return periods and the event magnitudes were logarithmically transformed. A regression was then fitted to the data with return period greater than or equal to 10 years, with the exception of the topranked outlier. Ordinary least squares (OLS) regression assumes error in only one variable, yet for flood frequency data there is uncertainty in both the discharge (introduced through gauging error or rating curve extrapolation) and in the return period assigned (due to selection of a particular plotting position formula, and to residual uncertainty as to the 'true' retum period). Hence, this is a Model II regression problem, and reduced major axis (RMA) regression is deemed more robust for

POWER-LAW EXTREME FLOOD FREQUENCY

145

Table 1. Return periods estimated by the Log Pearson III (LP3) model for observed extreme events at 50 US

river gauging stations (data courtesy of the United States Geological Survey (USGS), downloadable from the following website: http: / /waterdata.usgs.gov/nwis/sw). LP3 return periods were calculated using the UK Flood Estimation Handbook (FEH: Institute of Hydrology 1999) commercial software package 'WINFAP' US hydrologic region Alaska California California California Great Lakes Hawaii Hawaii Hawaii Lower Colorado Lower Colorado Mid Atlantic Mid Atlantic Mid Atlantic Mid Atlantic Mid Atlantic Missouri Missouri Missouri Missouri Missouri Missouri Missouri New England New England New England New England New England Ohio Ohio Ohio Ohio Pacific North West Pacific North West Pacific North West Red-Souri South Atlantic South Atlantic South Atlantic South Atlantic South Atlantic South Atlantic Tennessee Tennessee Tennessee Texas Texas Texas Upper Colorado Upper Mississippi Upper Mississippi Upper Mississippi

River Salcha Consumnes Deep C Eel Auglaize Honopou Kalihi Wailua San Pedro Santa Cruz Bushkill Missiquoi Raritan Creek Raystone Branch, Juaniata River Swatara Creek Cannonball Creek Elkhorn Little Missouri Marias Missouri South Boulder Creek Ten Mile Creek Mattawamkeag Piscataquis White Connecticut Quaboag Cheat Great Miami Mad Redbank Creek Grande Ronde Sandy White Salmon Wild Rice Bogue Chitto Broad Fishing Creek Linville Lumber Tombagee Clinch French Broad Powell Frio Llano Mora White Rocks Big Muddy Jump Mississippi

Largest event (m 3 s-a)

USGS gauge no.

Years of record

15484000 11335000 10260500 11477000 4191500 16587000 16229000 16060000 9471000 9482500 1439500 4293500 1400500 1562000

51 95 82 90 67 89 85 88 84 82 94 84 90 90

2.75 2.63 1.32 2.13 3.40 1.62 3.51 2.47 2.77 1.49 6.62 1.27 2.20 2.28

x x • • x x • x x x x x x •

103 103 103 104 103 102 102 103 103 103 102 103 103 103

4.04 x 1012 2.50 x 103 6.36 x 101 n/a n/a 2.87 x 102 2.14 x 103 n/a 5.35 x 104 3.22 • 103 1.97 • 105 2.03 • 107 3.88 x 104 4.44 x 104

1573000 6354000 6800500 6337000 6099500 6090800 6729500 6062500 1030500 1031500 1144000 1190070 1176000 3069500 3261500 3269500 3032500 13319000 14137000 14123500 5062500 2491500 2191300 2083000 2138500 2134500 2441500 3528000 3451500 3532000 8205500 8150000 7216500 9299500 5597000 5362000 5474500

84 87 81 68 96 61 62 86 100 100 86 102 89 89 89 62 96 80 90 81 81 80 103 88 80 74 89 82 107 68 86 85 73 84 60 86 123

1.89 2.68 2.83 3.11 6.82 3.96 2.09 9.31 1.32 1.06 3.40 8.86 3.62 4.81 1.25 1.57 1.88 3.99 1.74 1.28 5.75 3.54 1.98 8.52 1.12 7.08 7.58 2.78 3.11 1.68 6.51 9.03 3.96 1.31 1.21 1.30 1.26

x x • x x x x x x x x x x x • x x • x x • x x x x x • x x x x x x x x x x

103 103 103 103 103 103 102 101 103 103 103 103 102 103 103 103 103 102 103 103 102 103 103 102 103 102 103 103 103 103 103 103 102 102 103 103 104

8.71 1.04 9.12 5.72 5.60 5.38 4.34

n/a indicates the return period estimated by the model was so large as to be incalculable by the software.

LP3 return period (years)

x 10 l~ x 10 a~ x 102 • 102 x 103 x 105 x 105 n/a 8.58 x 104 3.00 x 103 2.86 x 10 x~ 1.56 x 104 5.66 x 103 3.16 x 103 6.13 x 104 7.24 x 103 6.73 x 103 3.54 x 104 1.84 x 103 4.36 x 107 3.20 x 103 4.99 x 104 3.02 x 103 1.09 • 103 9.16 x 102 2.75 • 103 6.11 x 102 1.00 • 104 1.30 x 106 4.83 x 103 2.86 x 105 6.49 x 101 2.21 x 103 n/a n/a n/a n/a

R. KIDSON ETAL.

146

Table 2. Details for 50 US river gauging records utilised in this study* Power law

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

LP3

Return % period Discharge (years) Probability error

Return period (years)

Probability

River

s.d.

Ratio

% Discharge error

Salcha Consumnes Deep C Eel Auglaize Honopou Kalihi Wailua San Pedro Santa Cruz Bushkill Missiquoi Raritan Creek Raystone Branch, Juaniata River Swatara Creek Cannonball Creek Elkhorn Little Missouri Marias Missouri South Boulder Creek Ten Mile Creek Mattawamkeag Piscataquis White Connecticut Quaboag Cheat Great Miami Mad Redbank Creek Grande Ronde Sandy White Salmon Wild Rice Bogue Chitto Broad Fishing Creek Linville Lumber Tombagee Clinch French Broad Powell Frio Llano Mora White Rocks Big Muddy Jump Mississippi

5.75 5.34 4.74 4.89 5.93 5.82 5.58 5.44 7.82 6.15 8.50 6.89 6.67 6.25

2.61 1.31 1.23 1.39 2.29 1.88 1.90 1.87 3.16 1.41 3.21 2.12 2.14 1.76

-46.6 -2.6 5.6 -20.0 -48.9 -32.7 -29.8 -32.5 -51.6 9.2 -62.8 -44.2 -40.8 -8.5

250 101 74 196 2945 317 299 463 369 74 1505 1205 880 108

0.20 0.94 1.10 0.46 0.02 0.28 0.28 0.19 0.23 1.11 0.06 0.07 0.10 0.84

-60.7 -34.8 10.2 -51.6 -58.1 -31.4 -42.8 -48.1 -70.8 -52.9 -72.4 -56.1 -51.9 -52.6

4.04 x 1012 2.50 x 103 6.36 x 101 n/a n/a 2.87 x 102 2.14 • 103 n/a 5.35 x 104 3.22 x 103 1.97 x 105 2.03 x 107 3.88 x 104 4.44 • 104

1.26 x 10 -11 3.80 x 10 -2 1.29 • 10~ n/a n/a 3.10 x 10 - t 3.97 x l0 -2 n/a 1.57 x 10 -3 2.55 • l0 -2 4.77 x 10 -4 4.14 • 10 -6 2.32 x 10 -3 2.03 x 10 -3

7.51 7.62 5.67 5.78 8.99 5.87 7.11

2.64 3.05 1.99 1.83 3.18 2.13 3.12

-48.4 -54.8 -41.1 -8.4 -53.4 -47.1 -56.0

524 594 919 79 248 1261 264

0.16 0.15 0.09 0.86 0.39 0.05 0.24

-67.4 -69.5 -39.8 -60.1 -80.4 -54.8 -75.6

8.71 1.04 9.12 5.72 5.60 5.38 4.34

9.64 8.37 8.88 1.19 1.71 1.13 1.43

8.30 5.10 5.45 7.32 5.38 7.49 6.80 6.71 5.71 6.61 5.18 4.76 7.73 6.49 5.14 5.51 5.78 6.14 5.20 5.15 5.42 7.98 5.20 7.95 5.54 6.90 5.04 5.38 6.77 3.88

3.31 1.59 1.64 2.52 1.25 1.51 1.89 2.13 1.82 1.88 1.59 1.28 2.95 2.03 1.87 1.62 1.48 2.12 1.87 1.38 1.73 2.58 1.80 2.79 2.02 2.27 1.69 1.64 2.20 1.24

-54.1 -28.9 -26.4 -48.6 -7.4 -0.0 -13.7 -43.3 -23.9 -23.6 -18.1 0.2 -56.7 -25.6 -33.4 -22.3 -20.6 -25.2 -29.1 -16.7 -28.1 -54.9 -37.4 -25.8 -34.8 -17.2 -31.6 -15.7 -47.8 -5.9

324 1130 401 741 131 90 117 756 110 188 152 90 654 140 349 232 159 155 276 163 286 2836 1365 121 378 94 451 83 1088 172

0.27 0.09 0.25 0.12 0.78 0.99 0.76 0.12 0.57 0.51 0.53 1.00 0.12 0.58 0.23 0.44 0.55 0.52 0.27 0.55 0.29 0.04 0.05 0.71 0.23 0.78 0.19 0.72 0.08 0.72

-80.5 -33.9 -36.9 -65.0 -36.2 -67.1 -53.0 -55.2 -53.9 -49.4 -41.7 -32.0 -71.1 -54.5 -40.2 -37.0 -41.3 -46.3 -40.1 -32.9 -40.7 -62.8 -41.2 -76.1 14.0 -64.6 -41.7 -59.3 -59.2 -42.4

n/a 8.58 x 104 3.00 x 103 2.86 x 10 m 1.56 x 104 5.66 x 103 3.16 x 103 6.13 • 104 7.24 x 103 6.73 x 103 3.54 • 104 1.84 • 103 4.36 x 107 3.20 x 103 4.99 x 104 3.02 x 103 1.09 x 103 9.16 x 102 2.75 x 103 6.11 x 102 1.00 x 104 1.30 x 106 4.83 x 103 2.86 x 105 6.49 x 101 2.21 x 103 n/a n/a n/a n/a

• x x • x • x

101~ 1010 102 102 103 105 105

1.17 3.34 3.01 6.54 1.57 2.81 1.45 8.57 1.43 2.26 4.88 1.86 2.53 1.60 3.41 8.04 8.74 2.70 1.46 8.20 8.25 1.41 3.01 1.31 3.31

x x x x • x •

10 - l ~ 10 -9 10 -2 10 -1 10 -2 10 -4 10 -4

n/a x 10 -3 x 10 -2 x 10 -9 x 10 -3 x 10 -2 x 10 -2 x 10 -3 x 10 -3 x 10 -2 • 10 -3 x 10 -2 x 10 -6 x 10 -2 x 10 -3 x 10 -2 x 10 -2 x 10 -2 x 10 -2 x 10 -1 x 10 -3 x 10 -5 x 10 -2 x l0 -4 x 10 ~ x 10 -2 n/a n/a n/a n/a

*Success in retrodiction of the largest event on record for two models fitted to the remainder of the data, the Power Law and the Log Pearson III (LP3), are shown. 's.d.' is the number of standard deviations of the most extreme event from the mean. 'Ratio' is the ratio of the extreme event to the next-ranking event on record. 'Probability' represents the probability of observing an event of the predicted return period during the actual period of record, n/a indicates the return period estimated by the model was so large as to be incalculable by the software.

POWER-LAW EXTREME FLOOD FREQUENCY flood frequency data, as it incorporates error in both variables. R M A regression was conducted with a specialist package (Bohonak 2004) available as freeware; a linear model was applied with gradient and intercept terms calculated according to Sokal & Rohlf (1981); the regression equation is

147

of the event at the observed outlier return period, and the return period of an event of the observed outlier magnitude. These results are also shown in Table 2.

Results a = ~ -

(3)

b2

where a is the RMA intercept, and b is the R M A gradient. An R M A regression was fitted to all the events (with the exception of the outlier) with a return period greater than or equal to 10 years. As the average record length of the rivers used was 84 years, a mean number of eight data points was available for regression. Into this regression was entered the return period of the observed outlier event, yielding an estimate of the magnitude of this event. The percentage error between this estimate and the actual magnitude observed was calculated. The regression was also used to generate the corollary: the observed magnitude of the outlier event was entered to estimate the return period. These results, and the details of each of the records, are contained in Table 2. Figure 3 plots an example of this procedure for the Cheat River (West Virginia), which also appears in Figure 1. An LP3 distribution with the USA Water Resources Council skew coefficient (WRC 1981) was also fitted to each of the gauging records. This was performed with the commercial FFA software produced by the U K Flood Estimation Handbook (FEH; Institute of Hydrology 1999). The model was fitted to a l l events except the largest one, because this is the normal basis for fitting this type of flood frequency distribution. The model was then used to predict both the magnitude

The results in Table 2 show that the average percentage error of estimation of the outlier event magnitude using the PL approach was 30.5%, and that there was a systematic tendency to underestimate the event magnitude. This represents a notably more accurate estimate than the LP3 model, which averaged a 50.4% underestimation error; in only three cases did the LP3 model make a more accurate retrodiction of discharge than the PL. These results are presented graphically in Figure 4. A more demonstrative divergence in the two models is apparent from the return period estimates of each (Table 2 and Fig. 5). The PL model produces far more conservative return period estimates than does the LP3 for the observed outlier events. As noted in the Introduction, the LP3 model often produces enormous (and in eight of the fifty cases incalculable) return period estimates that are unrealistic. It should be noted that the flood frequency analyst must consider two factors in relation to the top-ranked flood in a series. The first is the discharge for that return period estimated by the assumed model. The second is the probability that the highest-ranking flood event has an underestimated return period. This is occasioned by the fact that in, for example, an 80-year gauged

1104. 10 4 I []

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f

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[] Extreme Flood --RMA Regression

I b1 Return period (years)

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Fig. 3. Linear RMA regression fitted to annual floods with return period >_10 years in log-log space for the Cheat River, yielding a Power Law relation with which the top-ranked event may be retrodicted.

lO~

i62

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l&

Maximum observed discharge (m3 s - l )

Fig. 4. Graphical comparison of the PL and LP3 discharge predictions for the largest event on record, compared to the observed event, for the 50 rivers studied. Also shown is the one-to-one line for reference. In three cases the LP3 produced a closer estimate than the PL, but in only one of these cases was the difference between the models significant.

R. KIDSON ETAL.

148 '~ 1014! I0'~ 101% "~ 1011.

the models for two of these (Elkhorn River and Honopou Stream). This leaves only one record, that for the Llano River in Texas, where the LP3 model significantly outperformed the PL. This record is distinguished by having the largest coefficient of variation in annual flows of the entire data set. The literature has traditionally interpreted outlier events in one of two ways:

10 lo-

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60 80 1 O0 120 Observed (Weibull) return period (years)

Fig. 5. Comparison of the return period estimates of the PL and LP3 models against the observed (Weibull) return period. The one-to-one line is shown for reference; the PL model produces estimates that are far closer to those observed relative to the LP3 model. This figure omits the eight cases where the LP3 return period was so large as to be incalculable with the software. record, there is an 80% chance of including the largest event in 100 years, but the Weibull plotting position formula yields a return period of only 81 years. Applying the statistical convention of the 95% confidence interval, it is possible to calculate simply, for each gauging record, the probability of inclusion of the observed outlier event in the given record. This is n/T

(4)

where n is the record length and T is the estimated return period. Where this probability is less than 5%, it is concluded that it is unlikely that such an event would be incorporated in the record due to chance alone; hence this constitutes evidence to reject the model that generated the return period. When this simple scheme is applied to the return period results of both the PL and the LP3 models, it is seen that for the PL model, only four of the fifty cases exhibit a return period outside the 95% confidence interval. By contrast, for the LP3 model only seven cases fit within the confidence interval. If the stricter 99% threshold is applied, all of the PL estimates fall within and 29 of the LP3 estimates fall without. Statistically, the PL model represents the more plausible one for outlier event prediction. These results are noteworthy for the large range of catchment sizes (7-308,210 km2), event magnitudes (93-21,282m3s -1) and climatic regions (Alaska to Texas to Hawaii) over which this simple PL model provided superior estimations. Of the three cases where the LP3 model produced the more accurate retrodiction, there was only a marginal difference (i.e. less than 2%) between

(2)

It assumes that outlier events are derived from a separate process/population, and hence are not reconcilable within the same modelling framework. In this case, the 'outlier' events are excluded from the model-fitting process (e.g. Chow 1964; Viessman & Lewis 1996); It assumes that all events are derived from the same distribution, and accepts the resulting decrease in robustness of parameter estimation for conventional distributions occasioned by the inclusion of the outlier (e.g. Hawkins 1980; Siyi 1987).

If position (1) is assumed, it may be argued that the LP3 (and other similar models) constitute useful representations of the bulk of the data, notwithstanding the apparent support of the PL model for outlier event prediction. It might also be argued that the support demonstrated in Table 2 for the PL model is purely coincidental. For these reasons, it was desired to test the generality of the PL model against data records that do not exhibit obvious outlier events. For this analysis, a number of long-term rainfall records from the UK was utilized. These data were selected as (1) rainfall represents the hydrological variable of least skew (Kazmann 1965), and is thus unlikely to support a model based on a bias towards a heavy tailed distribution, which is the second conventional interpretation of records exhibiting an outlier event; (2) the UK possesses some lengthy rainfall records; (3) rainfall records are arguably subject to less measurement error than river discharge; and (4) the UK represents a relatively benign climate not prone to obvious climatic extremes. These reasons mean that these data constitute an appropriate 'null hypothesis' data set against which to test the PL model, as they are unlikely to support a model only capable of outlier prediction, or a model whose predictive capacity was coincidental in the case of the US river data. Total annual rainfall data were assembled from 12 long-term gauging stations in the UK. These data were obtained from three sources: the UK Meteorological Office website (UKMO 2005), the Radcliffe Meteorological Observatory (RMO 2005), and the long-term rainfall record maintained at Royston (Iceni) Weather Station (RWS 2005). A similar analytical method was applied to these

POWER-LAW EXTREME FLOOD FREQUENCY data: the events were ranked, observed (Weibull) return periods and annual totals were logarithmically transformed, and an RMA regression was fitted to all totals with a return period greater than or equal to 10 years (excluding the top-ranked annual total). The total rainfall and return period estimations for this PL model were calculated. For comparison, three conventional distributions were also fitted to the full data sets (also excluding the top-ranked annual total) in each case: the Generalised Logistic (GL), fitted with the L-moments method; the Generalised Extreme Value (GEV), fitted with the method of maximum likelihood; and the LP3, fitted with the WRC skew coefficient. The percentage error in the annual total estimations from each of these models was calculated, and the results are presented in Table 3. The standard deviations and ratios of top-ranked event to the next on record are modest for these records relative to those of the river discharge data in Table 2. Table 3 shows that the PL model best retrodicts the top-ranked annual total rainfall in six of the twelve cases, with the GL model best retrodicting three stations, the GEV two stations and the LP3 one station. The PL model also represents the best (i.e. lowest) percentage error as averaged across the 12 stations, with an average estimation error of 4.4%, compared to the GL (4.9%), GEV (7.5%) and LP3 (6.9%). These results indicate that the PL model performs well in the context of rainfall data that do not exhibit outlier events. This evidence supports the notion that PL models can be applied outside the context of outlier floods. It challenges the position that outlier events cannot be modelled within the same framework as the remaining members of a record (regardless of generating process); and suggests a more accurate alternative to conventional probability distributions. Moreover, the repeatability of these results across a large number of sites encourages the view that outlier hydrological events could be considered to represent a generalized pattern, rather than being unique, particular exceptions in individual cases.

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Discussion: goodness-of-fit and series length Conventionally, the utility of FFA models is judged according to goodness-of-fit of the predicted values from the model against the entirety of the observed data points. This approach is flawed for a number of reasons: (1) it assumes the observed data are precisely measured (Hall & Anderson 2002); (2) such tests are not sufficiently sensitive to differentiate competing models (Ramachandra Rao & Hamed 2000); (3) subjectivity may be introduced through the choice of different plotting position formulae (Stedinger et al. 1993), or the inclusion of different

O

~E

150

R. KIDSON ETAL.

portions of the data on which to conduct the test; and (4) different goodness-of-fit tests favour different models, in FFA as elsewhere (Lane & Richards 2001). This paper utilizes an alternative form of model assessment, reserving the top-ranked event from inclusion in the model-fitting process in order to test the model against this data point. This approach has obvious utility where outlier event prediction is the objective. However, this does not solve the problem of (1) above, and it can be seen that regression-fitting to a small number of events (i.e. those with a return period > 10 years; on average eight data points, for the records in this study) produces a gradient that is sensitive to error in the position of the included data points. There is considerable scope, therefore, for sensitivity analyses on this particular issue. Although in principle it may also be possible to exclude two 'outlier' events and conduct the same regression analysis, this must be balanced by the loss of information entailed by a reduction in the number of points available for regression. These results indicate that good predictions are made on the most extreme event of record, even when there is a relatively small size difference (see 'ratio' column in Table 2) between this and the next-ranked event. The results presented in Table 2 must also be considered in the context of measurement error in discharge estimates, particularly for outlier events (which are often estimated by extrapolation from a rating curve); the literature suggests that 20% error in measurement is not uncommon (Botma & Struyk 1970; Potter & Walker 1981, 1985; Cong & Xu 1987; Cook 1987; Quick 1991; Kuczera 1992, 1996). This point is made in order to reinforce the conclusion that hydrologists need to be pragmatic and realistic in their expectations of attainable prediction accuracy. This attitude is especially helpful when the assumption of emergent simple scaling is made, as this necessarily implies a local approximation to scaling behaviour. Of course, a short hydrological record of annual flood maxima may not exhibit an outlier event (as defined here), and this (in addition to the paucity of data points with return period > 10 years) may be interpreted as a weakness of the PL model developed here. However, it is perhaps risky to infer from absence that the PL model is not applicable to a particular locality; there is a reduced probability of outlier event expression in a short record, and this poses a diagnostic dilemma. We should perhaps recall the words of Oreskes & Belitz (2001; p. 28) and err on the side of precaution: Lacking adequate basis for analysis, modellers may leave out extremely low probability events. The small probability of the event discourages both its analysis and its incorporation. Yet

rare events do occur, and largely because they are unanticipated their impactis generallynegative.The omissionof verylow probability events exaggeratesour capacity to make accurate predictions and biases our modelsin an optimisticway. This raises the serious question of which approach is appropriate for stations with a short gauging record. Although a record length of 50 years yields five plotting positions with a return period greater than 10 years, a record length less than this yields very few points with return periods > 1 0 years on which to base a reliable regression. It should be noted that the real objective is to identify the phase transition threshold, and although this may well be observed with low n for some time series, the probability of manifestation in a short record is reduced. For this reason, it is likely to be inadequate simply to fit a regression to the tail of a time series regardless of its (short) length. There may be some scope for more sophisticated means to identify the phase transition (e.g. hill-climbing algorithms), although there is reason to be sceptical when sophisticated mathematical tools are employed in order to extract 'information' from limited data, rather than simply acknowledging lack of data. An alternative approach may be to use the peaks-over-threshold method, which may result in multiple events per year, and allows return periods of less than a year to be calculated. However, there is no guarantee that this will increase the number of events above a phase transition. In short, therefore, predictive (and explanatory) models for magnitude/frequency relations are less reliable when record length is short, a truism frequently acknowledged in the literature. Nevertheless, educated estimates may still be feasible, and Figure 6 illustrates a synthetically generated magnitude/frequency plot of 50 peak annual flood events, sampled randomly from a lognormal distribution. Recalling the annual series curvature issue enforced by the fact that return periods must exceed unity, it is possible that the artificial curvature evident in Figure 6 encourages the fitting of conventional concave-down distributions. The grey shaded region in Figure 6 illustrates the possible plotting space of larger (yet to be recorded) flood events. An objective procedure even as simple as fitting a regression to the entire annual series data with the exception of outlier events (without, of course, implying a causal relation as in classical regression, and notwithstanding bias attributable to residual serial correlation) may yield predictions within this outlier event space that could at least be of the correct order-ofmagnitude. In fact, this procedure was applied with good success to the data sets treated in this study, yielding more accurate predictions of the

POWER-LAW EXTREME FLOOD FREQUENCY 10 6b~ 10 5.

iil i!:!OutlierEventSpace

~ 10 4_~ 103 -m

,,,.

/

f

.= 10 0. . . . .

i

10 ~

101

....

i

10 2

10 3

Return period (years)

Fig. 6. A typical magnitude/frequency curve for peak annual discharges, plotted in double-logarithmic space. Possible plotting space for outlier events generated from an (unknown) self-similar process is shown in grey shading.

observed outlier events than conventional distributions (GL, GEV, LP3). The critical difference between such a method and these conventional distributions is that the former assumes (1) straight line, rather than concave-down scaling in log-log space, and (2) anticipation of a departure in plotting position of an outlier event relative to the remainder of the data. The approach is limited, of course, by lack of knowledge of both the phase transition threshold, and the gradient of the PL relation. However, estimation could be further refined by research investigating the relationship of the various separate factors influencing the PL gradient (catchment size, slope, shape, and climate), and by regionalization studies as in conventional FFA. Such an approach could inform decisions about where in the upper range of events the phase transition occurs, and therefore how many plotted events should be used to fit the extrapolation of the upper range of floods in Figure 6 (the segment including the highest five events, or that including the highest dozen). Also significant is that no a priori assumptions regarding separate generating mechanisms for outlier floods have thus far been made in order to yield good predictive success. Mixed flood generating mechanisms have long been recognized and partitioned in modelling practice (e.g. Waylen & Woo 1982; Waylen 1985), although this approach is only possible when there are a sufficient number of events from an individual mechanism to warrant separate modelling. Where this is the case, the methodology described here could be usefully applied to the separate mechanisms in order to refine predictions. For cases where generating mechanisms are obscure or not sufficiently

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represented in the record, these error components are subsumed by the assumption of simple scaling; although open to more refinement, this is apparently not a confounding assumption over limited scales of interest. There is scope for additional research in these areas. There is also scope to examine mechanism-independent processes that are a simple function of the large size of an event (regardless of source), for example, the characteristics of extreme precipitation events, catchment runoff processes and exceedance thresholds for saturation excess overland flow, and differential routing distances dependent on the location of rainfall and the structure of the drainage network. These physical processes could be profitably explored in a rainfall-runoff modelling framework, and may provide physical explanations for theoretically postulated selfsimilar behaviour. Some approaches in this direction are made by Jothityangkoon & Sivapalan (2003).

Conclusions This paper has compared a simple Power Law (PL) model with conventional distributions for the prediction of outlier flood events. The assumption of emergent simple scaling was made, which permitted the fitting of a single PL relation over a limited scale range. Fifty river gauging records exhibiting a high outlier event were studied. Based on the empirical results of Potter (1958) the 10-year return period was applied as a uniform threshold above which straight-line scaling in log-log space was hypothesized. For each record, an RMA linear regression was fitted to all events with a return period >10 years, excluding the outlier event. The regression was then used to retrodict the discharge and return period of the outlier event. Comparison with the retrodictions made by the Log Pearson III distribution showed the clear superiority of the PL model for outlier event estimation. In order to test the generality of the model, twelve long-term rainfall records from the UK, exhibiting no outlier events, were similarly studied, and the PL model was found to be the most accurate model in predicting the largest event on record compared to three common distributions. It is concluded that the PL model represents a promising alternative to conventional approaches to outlier event prediction, from both an empirical and theoretical perspective. In particular, it offers a meaningful avenue through which to consider both outlier and the bulk of floods within the same framework. The main consequence of the employment of the PL model over alternatives is a far more conservative estimate of the return

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period of large events, which has particular significance for managing extreme floods.

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POWER-LAW EXTREME FLOOD FREQUENCY MANDELBROT, B. B. & WALLIS, J. R. 1969. Some long-run properties of geophysical records. Water Resources Research, 5, 321-340. ORESKES, N. & BELITZ, K. 2001. Philosophical issues in model assessment. In: BATES, P. D. (ed.) Model Validation: Perspectives in Hydrological Science. John Wiley and Sons, Chichester, 23-42. POTTER, W. D. 1958. Upper and lower frequency curves for peak rates of runoff. Transactions of the American Geophysical Union, 39, 100-105. POTTER, K. W. & WALKER, J. F. 1981. A model of discontinuous measurement error and its effects on the probability distribution of flood discharge measurements. Water Resources Research, 17, 1505-1509. POTTER, K. W. & WALKER, J. F. 1985. An empirical study of flood measurement error. Water Resources Research, 21,403-406. POTTER, W. D., STOVICEK, F. K. & WOO, D. C. 1968. Flood frequency and channel cross section of a small, natural stream. Bulletin of the International Association of Scientific Hydrology, 13, 66-76. QUtCK, M. C. 1991. Reliability of flood discharge estimates. Canadian Journal of Civil Engineering, 18, 624-630. RAMACHANDRARAt, A. & HAMED, K. H. 2000. Flood Frequency Analysis. CRC Press, Boca Raton. RMO 2005. Radcliffe Meteorological Station. World Wide Web Address: http://www.geog.ox.ac.uk/ research/rms/. [Accessed 1 July 2006]. RODRIGUEZ--ITURBE, I. & RINALDO, A. 1997. Fractal River Basins: Chance and Self-Organisation. Cambridge University Press, Cambridge. RWS 2005. Royston (Iceni) Weather Station. World Wide Web Address: http://www.iceni. org.uk/index/rain.htm. [Accessed 1 July 2006]. SCHEIDEGGER, A. E. 1997. Complexity theory of natural disasters; boundaries of self-structured domains. Natural Hazards, 16, 103-112. SCHERTZER, D., LOVEJOY, S., LAVALLEE, D. & SCHMITT, F. 1991. Universal hard multifractal turbulence, theory and observations. In: SAGDEEV, R. Z., FRISCH, U., HUSSAIN, F., MOISEEV, S. & EROKHIN,N. S. (eds) Nonlinear Dynamics of Structures. World Scientific, New Jersey, 213-235. SCHERTZER, D., LOVEJOY, S. & LAVALLEE, D. 1993. Generic multifractal phase transitions and self

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Models, data and mechanisms: quantifying wildfire regimes J A M E S D. A. M I L L I N G T O N t, G E O R G E L. W. P E R R Y 2 & B R U C E D. M A L A M U D 1

XEnvironmental Monitoring and Modelling Research Group, Department of Geography, King's College London, Strand, London WC2R 2LS, UK (e-mail: james, millington @kcl. ac. uk; bruce @malamud, corn) 2School of Geography and Environmental Science, University of Auckland, Private Bag 92019, Auckland, New Zealand (e-mail: george.perry @auckland, ac. nz) Abstract: The quantification of wildfire regimes, especially the relationship between the frequency with which events occur and their size, is of particular interest to both ecologists and wildfire managers. Recent studies in cellular automata (CA) and the fractal nature of the frequency-area relationship they produce has led some authors to ask whether the power-law frequency-area statistics seen in the CA might also be present in empirical wildfire data. Here, we outline the history of the debate regarding the statistical wildfire frequency-area models suggested by the CA and their confrontation with empirical data. In particular, the extent to which the utility of these approaches is dependent on being placed in the context of self-organized criticality (SOC) is examined. We also consider some of the other heavy-tailed statistical distributions used to describe these data. Taking a broadly ecological perspective we suggest that this debate needs to take more interest in the mechanisms underlying the observed power-law (or other) statistics. From this perspective, future studies utilizing the techniques associated with CA and statistical physics will be better able to contribute to the understanding of ecological processes and systems.

In many regions of the world, wildfires are common and are considered an integral component of ecosystem functioning. However, wildfires also pose a threat to humans, their activities and livelihoods, and repeated fires can negatively affect ecosystem functioning (Bond & van Wilgen 1996). Thus, understanding and managing the relationships between wildfires, ecological systems and human activity is important. The combination of the timing, frequency and magnitude of all disturbances occurring in a given region is known as the 'disturbance regime'. Recently, much research has considered one particular aspect of the disturbance regime: the frequency-area distribution of wildfires in a given area. Here, we will focus on disturbance by wildfires. Examination of these statistics in the context of wildfire activity is not new (e.g. Minnich 1983; Baker 1989; Strauss et al. 1989, among many others), but recently there has been considerable debate regarding the 'heavy-tailed' (i.e. the tail decreases at a relatively slow rate) nature of these frequency-area distributions. One specific class of heavy-tailed distribution is a power-law (fractal) where the frequency-area distribution has no inherent scale (it is scale-invariant). The presence of such scaling relationships has been noted widely in many features of biological and ecological systems (e.g. Brown et al. 2002). In the wildfire literature, discussion has particularly addressed From: CELLO,G. & MALAMUD,B. D. (eds) 2006.

whether these heavy-tailed frequency-area distributions are power-law in nature, and what the implications of such a power-law distribution might be. Much of the present debate on the heavy-tailed nature of 'real' wildfire areas is the result of research in the early 1990s, where simple 'forestfire' cellular-automata (CA) models were found to produce power-law size frequency distribution - a characteristic linked with self-organized criticality (SOC) (Bak et al. 1990; Drossel & Schwabl 1992; C l a r e t al. 1996). Malamud et al. (1998) then produced the first detailed research showing that both the forest-fire CA model and 'real-world' wildfires exhibit robust power-law frequency-area distributions. Since then, other authors have presented data and analyses with the aim of variously confirming or refuting the assertion that real-world wildfire frequency-area distributions follow a power-law distribution (e.g. Ricotta et al. 1999, 2001; Cumming 2001; Ward et al. 2001; Reed & McKelvey 2002; Schoenberg et al. 2003). In this paper, we will examine the history and nature of this discussion, before suggesting what direction it might take, or be most useful to take in the future. We approach this topic from a broadly ecological perspective, emphasizing the need for consideration of the ecological (or otherwise) mechanisms driving observed wildfire frequency-area distributions. We will begin by

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examining the most recent papers in this area, before establishing the state of current research in this area and suggesting what avenues of future research on this topic might prove fruitful.

Ecological examination of wildfire frequency-area distributions Consideration of wildfire and other disturbance regimes (the spatio-temporal dynamics of recurrent disturbance events) has a long history in ecology. The dynamics of succession-disturbance in ecology is important when considering wildfires. Succession is the change in ecological community composition (in essence the relative abundance of the different species in the community) and function (the ways in which the abiotic and biotic components of the community are linked) through time. Disturbance is the disruption of an ecosystem, community, or species' populations by any relatively discrete event in space and time, with a resultant change in the physical environment (White & Pickett 1985). Until the 1950s and 1960s, ecologists' views on succession-disturbance dynamics were dominated by the perspective of Frederick Clements (1916, 1928, 1936). Clements' conceptualization of the community emphasized equilibrium and stability, as encapsulated by the 'balance of nature paradigm'; in this view disturbance events were seen as unnatural, as they moved the system away from its 'natural' equilibria (the so-called 'climax' condition). Ecosystem management conducted from this perspective, therefore, aimed to minimize disturbance events and their impacts, resulting in policies such as fire suppression. More recently, ecologists have accepted the fundamental importance of apparently random events, such as disturbance, in structuring ecosystems, and have adopted a more scale-sensitive, disequilibrial view (Wu & Loucks 1995; Perry 2002). With this shift has come increasing interest in characterizing the three key dimensions of the disturbance regime: size, frequency, and intensity. Recently, there has been some attention focused on determining whether large, infrequent disturbance have a qualitatively different effect than small, frequent ones (Romme et al. 1998; Turner et al. 1998). It is with this historical perspective in mind that we need to consider ecological approaches to quantifying wildfire regimes, in contrast to the model-based approaches we discuss later. The frequency of disturbance events is very important in terms of the evolution of the reproductive strategies that different species adopt (e.g. the number and size of offspring produced, the energy invested per reproductive event, and so on); these reproductive strategies are sometimes also known

as life history traits (Bond & Keeley 2005). For example, the optimal time after disturbance for a species to maximize seed storage (in either the crown or soil seedbank) will be influenced by the average time between wildfire events (Enright et al. 1998a, b). What constitutes a 'frequent' wildfire will vary from ecosystem to ecosystem, depending on factors such as rates of biomass production, the nature of other disturbance agents operating alongside fire (e.g. wind-throw) and regeneration rates. An intriguing body of ecological theory suggests, however, that intermediate disturbance frequencies will promote the highest levels of biodiversity (the 'intermediate disturbance hypothesis', Connell 1978). Early quantitative studies of the wildfire regime, conducted in the 1950s and 1960s, emphasized frequency - in essence an estimate of the probability distribution of survival or mortality from wildfire(s) (Johnson & Gutsell 1994). Early efforts (e.g. Spurr 1954) were often somewhat ad hoc studies of wildfire occurrence, and are perhaps better seen as wildfire 'history' studies. However, Heinselman (1973), in a seminal study, mapped the time-since-wildfireyear, on the basis of stand ages, in the Boundary Waters Canoe Area in Minesotta (USA). On the basis of this map Heinselmann estimated survivorship from wildfires in the landscape. Since the late 1970s, a number of statistical methods and distributions that might be suitable for describing wildfire frequency have been developed and applied, with much emphasis on the Weibull and negative exponential distributions (see Johnson & van Wagner 1985). These statistical models allow empirical assessment of relationships between spatio-temporal variation in wildfire frequency and other environmental factors (Johnson & Gutsell 1994). Considerable debate remains over the drivers of spatio-temporal variability in wildfire frequency (in particular the relative roles of weather v. fuels), and unravelling these patterns is a focus of current work (e.g. Bessie & Johnson 1995). Although sophisticated statistical tools are available for modelling fire frequency (e.g. Presiler et al. 2004; Reed & Johnson 2004), the stumblingblock is often collecting adequate empirical data to represent the processes and designing adequate sampling strategies for this data collection (Johnson & Gutsell 1994). Although much research effort has focused on the frequency component of the wildfire regime, other ecologists have considered the size (i.e. burned area, often equated with severity) component of the wildfire regime. As different ecosystems respond differently to wildfires, what constitutes a severe event will also vary (Moritz 1997). The diverse effects of wildfire suppression efforts have received considerable attention in this context.

QUANTIFYING WILDFIRE REGIMES Minnich (1983) compared the frequency-area distribution for regions in Southern California that had been subject to wildfire suppression, with that of regions in northern Baja California that had not. He found that in regions subject to suppression, large intense wildfires occurred (possibly larger than had occurred pre-suppression), and that total burned area was the same as in regions where wildfires were unsuppressed. Although subsequently there has been much debate concerning the existence and significance of this difference (e.g. Strauss et al. 1989; Chou et al. 1993; Keeley & Fotheringham 2001; Minnich 2001), this research stimulated interest and debate in the appropriate methods for characterizing and comparing empirical wildfire frequency-area distributions (see also Miyanishi & Johnson 2001; Ward et al. 2001; Bridge et al. 2005, for discussion regarding Ontario). The consensus appears to be that the majority of empirically observed wildfire size distributions are heavy-tailed (e.g. Malamud et al. 1998, 2005).

The forest-fire cellular automata model The forest-fire cellular automata model rose to prominence amidst the suite of models used by Per Bak and others to examine and propound the theory of self-organized criticality (SOC) in dynamical systems (Bak et al. 1987, 1988). Self-organized criticality was first presented by Bak et al. (1987) as the concept that dynamical systems order themselves naturally to a critical state regardless of initial conditions and independent of any exogenous driving force. Although the exact definition of SOC is often unclear, Turcotte (1999, p. 1380) suggests a working definition for SOC 'is that a system is in a state of self-organized criticality if a measure of the system fluctuates about a state of marginal stability'. Bak (1996) suggested that, at the critical state, small inputs to a system can cause events of any magnitude in intermittent periods of activity and that prediction of the size of a specific future event is impossible. The frequency-area distribution of events in this type of system will exhibit power-law (i.e. critical) behaviour (Bak & Tang 1989). Bak et al. (1987) first presented the concept of SOC using models of coupled-pendulums and, more famously, sandpiles (for further discussion on the sandpile model, see paper in this volume, Malamud & Turcotte 2006). Later, Bak et al. (1990) used a cellular automaton (CA) to model forest fires. Although there are many variations on this forest-fire model, details of the rules that define the mechanics of the simplest of these models can again be found in Malamud & Turcotte

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(2006); an example of the progression of this simple forest-fire model is given in Figure 1. Using the simplest version of the forest-fire CA model, Bak et al. (1990) and Drossel & Schwabl (1992), amongst others, found that uniformly injected energy ('trees') is dissipated ('burning trees') in a spatially self-similar (fractal) manner. The forestfire CA model was shown to self-organize itself to a state where fire sizes (patches of contiguous cells that burn in a single burning 'event') exhibited power-law frequency- area statistics. Similar models had previously been considered in the context of percolation theory (Albinet et al. 1986; Beer & Enting 1990, 1991). Site-percolation models are specified by a parameter, p, which defines the probability of a site in a lattice being occupied. At a critical point (Pc ~ 0.59275) clusters traversing the entire lattice (spanning clusters) form. Although these models are not self-organizing through time, they do show critical behaviour in the sense that the model shows quite different regimes of behaviour depending on the value of the tuning parameter. These percolation-based models have received less attention than the forest-fire CA model in the context of SOC, which rapidly attracted interest because of its apparent potential to contribute to the understanding of natural dynamical systems (they are, however, the basis of much research in statistical physics). Hergarten (2002) makes this same point with reference to a CA model, essentially the same as that used by Drossel & Schwabl (1992), proposed by Henley (1993). Turcotte (1999) provides a good comparison of the CA forest-fire model and site-percolation models. A non-cumulative frequency-area distribution is considered power-law if

f(A) ~ A -~

(1)

where f ( A ) is the frequency density, that is, the number of wildfires with burned area A (properly normalized to 'unit' bins), and/3 is a constant. Plotting the frequency densities against area in log-log space produces a straight fine with slope -/3. Power-law frequency-area behaviour has often been interpreted as a sign that a system is in an SOC state, as it suggests emergent global properties that have risen from simple local interactions (Solow 2005). As a result, the presence of powerlaw frequency-area statistics in the forest-fire CA m o d e l are suggestive of SOC behaviour, particularly because the properties are robust to the values of various forcing parameters and the values of the critical exponents of the model appear 'universal' (i.e. the frequency-area power-law exponent

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remains independent of any m o d e l parameters; Clar et al. 1994).

Based on the dynamics of these simple models, proponents of SOC suggested that it 'might be the underlying concept for temporal and spatial scaling' in dynamical systems (Bak et al. 1987, p. 384) and even that it might define H o w N a t u r e W o r k s (Bak 1996). With hindsight these claims seem overstated, particularly considering the way in which this modelling process proceeded. As

Hergarten (2002) points out, modelling in earth and environmental sciences often starts by observing a set of p h e n o m e n a and then proceeds to attempt to represent these p h e n o m e n a as accurately as possible in a modelling framework. However, these m u c h more abstract CA models were developed by statistical physicists with little regard to the actual processes they were representing, with the 'forest fire' label originally intended more as a metaphor rather than a claim of representation. As

QUANTIFYING WILDFIRE REGIMES generalized models of abstract systems these experiments are interesting, but whether their behaviour is present in the real world has been questioned (e.g. Reed & McKelvey 2002). There are of course many questions that remain, but two of interest to earth scientists and, in particular, ecologists, include (1) whether the patterns and behaviours observed in the CA models are found in nature? and (2) whether 'real-world' forests and their disturbance regimes are/show SOC?

Confronting models with data O b s e r v e d f r e q u e n c y - a r e a distributions in nature

Malamud et al. (1998) were the first to examine whether the power-law frequency-area distributions of model fires in the forest-fire CA model were also characteristic of 'real-world' wildfire regimes. Examining data sets from four study areas they found that wildfire frequency-area distributions followed a power law (see Table 1). In these four regions, spread around the globe and with widely varying environmental conditions, frequency- area power-law behaviour was observed over up to six orders of magnitude, with the powerlaw exponent/3 = 1.3-1.5. Malamud et al. (1998) also attempt to interpret some of the parameters in the forest-fire CA model in the context of observed fire dynamics and regimes. The 'sparking frequency' parameter (the frequency with which model fires are given the potential to start by a match) was directly compared to management strategies practised in Yellowstone National Park, and the implications of changes in the parameter values discussed with reference to real-world events. Although the exponents in the frequencyarea power-law relationship differ between the real world and model data (with the model data exhibiting consistently smaller/3 values), the suggestion by Malamud et al. (1998) was that wildfires could be quantified in nature by using the same frequency-area scaling relationship found in the forest-fire CA model (see also Turcotte 1999). The implication was that the ecological systems in which real wildfire regimes exist may potentially exhibit SOC behaviour in the same way as the CA models appear to. Malamud et al. (1998) were not the first to find power-law scaling in wildfire frequency-area statistics (e.g. Minnich 1983, as discussed above in the section on ecological studies). However, they were the first to compare wildfire regimes in nature and in the forest-fire CA model. Furthermore, the prominent location of this publication, allied with enthusiasm in much of the scientific

159

community at the time for SOC, meant that a flurry of similar studies making similar analyses of real wildfire regimes soon followed (see Table 1). The majority of these studies (including the one by Malamud et al. 1998) were hampered by low spatial and/or temporal resolution, with relatively few wildfire records. Recently, however, Malamud et al. (2005) have examined a much larger, high-resolution (spatial) data set detailing the burnt area, location and cause of ignition of 88,916 wildfires on United States Forest Service land across the conterminous USA for the period 1970-2000 (this is also discussed in Malamud & Turcotte 2006, in this volume). The large amount of data allowed Malamud et al. to examine different subregions of the conterminous USA, and compare them with each other. For each of the 18 regions examined, excellent power-law relationships were found between the non-cumulative number of wildfires and burned area, with / 3 = 1.30-1.81. Two examples are given in Figure 2, showing the two extremes of values of 13 obtained by the authors. In Figure 2a are presented the frequency-area statistics for 16,423 wildfires in the Subtropical ecoregion division (within the southeastern part of the USA) and in Figure 2b, 475 wildfires in the Mediterranean ecoregion division (within California, USA). In both cases, excellent correlations are obtained with the power-law relationship (1), with / 3 = 1 . 8 1 _ 0.07 (__+2 s.d.) for the Subtropical ecoregion and /3 = 1.30 _ 0.05 ( + 2 s.d.) for the Mediterranean ecoregion. The values of/3 for model fires in the forest-fire CA model are consistently lower than those of real-world wildfires. This indicates a reduced contribution to the wildfire regime of small wildfires, and a corresponding increased contribution of large wildfires, when comparing the real world with CA models (as observed by Malamud et al. 1998). Why there is such a consistent disparity is a question that needs to be addressed if links between the forest-fire CA model and real wildfire regimes are to be made. H e a v y tailed, but w h a t f l a v o u r ?

The large majority of studies have found heavytailed wildfire frequency-area distributions (Table 2), with the implication being that extreme events are perhaps not as extreme (or surprising) as they are often perceived to be (Katz et al. 2005). There is, however, considerable debate as to what type of probability distribution best describes these data. Empirically driven studies have used a range of heavy-tailed distributions, including the Weibull and the generalized Pareto among others, whereas studies arising from

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Fig. 2. Normalized frequency-area wildfire statistics for (a) Mediterranean and (b) Subtropical ecoregion divisions, for the period 1970-2000 (figure after Malamud et al. 2005). Shown (circles) are normalized frequency densitiesf(AF) (number of wildfires per 'unit bin' of 1 km 2, normalized by database length in years and USFS area within the ecoregion) plotted as a function of wildfire area AF. Also shown for both ecoregions is a solid line, the best least-squares fit to Eq. (1), with coefficient of determination r 2. Dashed lines represent lower/upper 95% confidence intervals, calculated from the standard error. Horizontal error bars are due to measurement and size binning of individual wildfires. Vertical error bars represent two standard deviations ( ___2 s.d.) of the normalized frequency densities j~(AF). exploration of the forest-fire CA model have emphasized power-law distributions (see Table 1). Some have queried whether 'true' power-laws would be expected in nature (e.g. Bolliger et al. 2003), and others whether observed wildfire regimes do actually follow a power-law (e.g. Reed & McKelvey 2002). Malamud et al. (2005) acknowledged that there will always be upper and lower cutoffs in nature for any power-law behaviour and that a true mathematical power-law (fractal) would be impossible in nature (see also Brown et al. 2002). There will always be a lower limit to what can be described as a wildfire, and

measurement accuracy of the smallest wildfires is problematic. At the upper bound both Reed & McKelvey (2002) and Malamud et al. (2005) cite topographic influences restricting wildfire spread and therefore putting a constraint on the largest possible wildfire. Such effects are analogous to the finite-grid size effect observed in the forestfire CA model. Schenk et al. (2000) showed the finite-grid effect results in a collapse of the fieq u e n c y - a r e a power-law scaling relationship when the correlation length (a measure of the radius of the largest tree cluster) becomes large relative to the size of the system (i.e. the size of the grid).

Table 2. Examples o f heavy-tailed wildfire frequency-area distributions suggested by recent studies* Nature of distribution

Author

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See Table 1 Baker (1989) Reed & McKelvey (2002)

Truncated power law

(a) Burroughs & Tebbens (2001) (b) Cumming (2001) (c) Schoenberg et al. (2003)

Study area (time period) [no. of wildfires examined] Minnesota (1727-1868) [not stated] (a) Sierra Nevada, California (1908-1992) [2536] (b) Nez Perce NF, Idaho (1870-1994) [1795] (c) Clearwater NF, Idaho (1910-1999) [884] (d) Yosemite NP, California (1930-1999) [3190] (e) N.E. Alberta (1961-1998) [5478] (f) Northwest Territories (1992-1999) [2544] (a) Australian Capital Territory (1926-1991) [298] (b) N.E. Alberta (1980-1993) [2898] (c) LA County, California (1950-2000) [548]

*These studies are based on empiricallyobserved, rather than model-derived,data and analyses. Although a variety of different distributions are suggested, all are heavy-tailedin nature and many suggest power-lawbehaviourover a limited range of magnitudes. NF: NationalForest; NP: National Park; LA: Los Angeles.

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As noted above (section on ecological studies), studies in the 1970s to the 1990s examined wildfire frequency-area relationships and suggested that they follow distributions other than the power law (e.g. Baker 1989). More recent studies have also suggested that wildfire distributions are not truly power-law, in the sense that they are not powerlaw across all event magnitudes (see Table 2). For example, Burroughs and Tebbens (2001) examined the same Australian Capital Territory (ACT) data used by Malamud et al. (1998) in a cumulative plot and proposed that a truncated power law offered a better fit. This then, is one criticism of the contention that real-world wildfire regimes show similar behaviour to those of the forest-fire model - the fact that real-world wildfires may not actually exhibit power-law (i.e. scale-invariant) behaviour in their frequency-area statistics. Reed & McKelvey (2002) specifically considered the question of whether power-law fire size distributions existed in real-world data as suggested by the presence of SOC behaviour. Examining six regions in North America (see Table 2), the authors suggested that the presence of power-law behaviour in wildfire size distributions was exaggerated, and that variations on the Weibull distribution provided the best fit to data. Reed & McKelvey (2002) did concede that power-law behaviour was in evidence over a limited range of wildfire sizes for some regions. The examples cited here (Table 2) consider distributions other than the power law provided better fits to empirical data. However, for all the regions studied, these alternative distributions are 'heavy-tailed', with many of them very closely related to the two-parameter power-law distribution, but with additional parameters. A number of these distributions still exhibit scale-invariance over some part of the range of wildfire sizes they examine. Discrepancies between these distributions are most prominent in the tails where infrequent, extreme events cause distortion. More generally, there are other distributions that may explain the power-law distributions observed; for example, the log-normal and exponential distributions look quite linear in log-log space, especially when the distribution extremes are 'veiled' (May 1975; Brown et al. 2002). The infrequent, but very large events, in the heavy tail of the wildfire size distributions are of most concern for fire and forestry managers, but it is also the area where the most uncertainty lies. Wildfire size distributions that inaccurately represent the upper tail are problematic for managers. However, it is often the case when dealing with probabilistic hazard forecasting that the uncertainty for the recurrence intervals for the largest events (e.g. earthquakes and floods) is very large, and managers 'make do' with what is available. Thus, we believe it is important that any

attempt to fit a specific heavy-tailed distribution to data is accompanied by error bars and a measure of the confidence of those fits (Fig. 2) so that the uncertainty by the hazard managers can be fairly addressed. In many studies error terms and measures of confidence are not provided. The debate regarding whether power-law frequency-area distributions and associated SOC-type behavionr, as found in the forest-fire CA model, are also found in nature remains open; in the next section, we discuss possible new directions for this effort.

Whichever flavour, what does this all mean for future research? Perspectives on SOC

The current state of research contests the presence of power-law behaviour in real wildfire regime frequency-area distributions as evidence of SOC-type mechanisms (Gisiger 2001; Frigg 2003; Solow 2005). It should be remembered that the FFCA, and other models of its type, is a metaphor for SOC behaviour rather an explicit representation of a specific system and its associated suite of processes. The spatially random recovery (re-growth) of trees is a weak assumption as seed dispersal processes will determine tree regeneration pattems (this type of recovery was simulated in a recent model of mussel-bed disturbance, Guichard et al. 2003). There are many other mechanisms by which power-law behaviour can be generated in nature using the presence of power laws to determine whether a system is SOC suffers from the problem of under-determination (e.g. Oreskes et al. 1994; Frigg 2003). For example, another mechanism that has been proposed to explain the presence of power-law size-frequency distribution in many systems is Highly Optimized Tolerance (HOT; Carlson and Doyle 1999, 2002; Doyle and Carlson 2000). HOT takes a rather different view of complex systems than SOC, focusing on 'designed' systems (whether engineered or subject to natural selection) that are optimized to be robust in the face of environmental uncertainty. In itself the simple presence of a power law is a very weak test of the presence of any specific generating mechanism, and the way investigation is currently being pursued does not improve on this. After all, how good does a power-law relationship have to be to show SOC behaviour? However, irrespective of the origins of power-law behaviour, and despite the criticisms levelled by some authors advocating altemative distributions with better fits to the data, the power law is currently the most parsimonious model available to describe wildfire frequencyarea distributions.

QUANTIFYING WILDFIRE REGIMES The implication that SOC behaviour demands true power-law behaviour over all magnitudes of events is, of course, in reality impossible. Only in an infinite system is this possible, but the world is finite. Therefore, not only is it no surprise that we do not find 'true' power-law behaviour in nature, but upper and lower cutoffs in any observed power-law behaviour are inevitable. However, possibly the biggest problem regarding the credibility of studies examining wildfire (or other 'natural') distributions and finding, or advocating, power laws, is the feeling that the researcher is simply fitting a line through their data with little regard for what this means or how this new finding might be used. In terms of interpreting observed distributions as the fingerprints of underlying process, differences between distributions (no matter how subtle) imply different generating mechanisms. As Brown et al. (2002, p. 622) comment 'In current applications of statistics to biological or ecological data, there is often an unfortunate tendency to be satisfied with the "model" or equation that gives a good fit. It is important, however, to consider the implications of the particular mathematical form of the equation'. Examining abstract systems (models) is a valid scientific pursuit in itself (as the forest-fire CA model, and so on, were initially used), but once we enter the realm of the actual, questions of 'Why?' and 'What use?' become of greater importance. Specifically, here we propose the questions 'Why is this system exhibiting power-law behaviour?' and 'To what use can this power-law nature be put?' should drive future research in this field. Mechanisms and causal processes

Recently, some authors have begun to examine why power laws are observed in both models and nature. For example, Reed & Hughes (2002) suggest that if stochastic processes growing in an exponential manner are 'killed' at random (the burning of trees in the forest-fire CA model), the distribution of this killed state will follow a power law in one or both tails. Yang (2004) suggests power-law behaviour in SOC systems is the result of a balance between competitive trends. Specifically, power-law behaviour occurs when the probability of a site being in an 'active' state (rather than 'inactive') at the next time step is close to 0.5 (i.e. in the forest-fire CA model, the probability of being a tree versus being empty due to burning). Attempts to link the theory and pattems observed in SOC-type models to observed ecological patterns and processes are also on the horizon. Pascual & Guichard (2005) highlight the differences between three types of criticality ('classical', 'self-organized' and 'robust') and their relevance to disturbance patterns observed in ecological systems. These authors

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suggest that systems with subtle variation in their relationships between disturbance and recovery show different types of critical behaviour. Greater consideration of the processes driving SOC-type, power-law behaviour is required and links to observed processes and pattem, such as that demonstrated by Pascual & Guichard (2005) should be welcomed. Despite the issues regarding wildfire frequencyarea distributions, and the spatial restrictions on power-law behaviour, Malamud et al. (2005) emphasize the usefulness of power-law distributions for describing and studying the drivers of wildfire regimes at regional and continental scales. By spatially disaggregating data into Bailey's (1995) ecoregions (geographic areas of similar climate, vegetation and soil), and normalizing frequency-area statistics by ecoregion area and number of years in each data set, differences between wildfire regimes could be compared and contrasted with reference to putative broad-scale environmental drivers. Results showed an apparent east-west gradient in the power-law exponent (/3 values) across the conterminous USA, over 18 different ecoregions, potentially due to forest fragmentation and/or human population densities. Holmes et al. (2004) used a similar methodology to examine differences in Florida wildfire regimes according to vegetation type ('flatwoods' v. 'swamp'). Malamud et al. (2005) suggested that examining relationships between past and current climates could aid understanding as to how wildfire frequency-area scaling might change under future modified climate conditions. This type of approach is advocated by Brown et al. (2002), who believe that improved analysis of empirical patterns is required (in particular they advocate looking for systematic deviations from self-similarity). F o r e c a s t i n g a n d p r e d i c t i o n v. e x p l a n a t i o n

Malamud et al. (2005) used their power-law frequency-area statistics to do probabilistic hazard analysis, where they calculated wildfire recurrence intervals - the average time between events of a given area or larger - for spatial 'areas' of 1000 km z in each of the ecoregions. A map of expected recurrence intervals for wildfire areas of 10 k m 2 o r greater was presented for the 18 ecoregions of the conterminous USA (Fig. 3). Examples of recurrence intervals found included 2 • 1 years ( • 2 s.d.) in the Mediterranean ecoregion and 203 -t- 99 years (_+ 2 s.d.) in the Warm Continental ecoregion. In other words, for the Mediterranean ecoregion, in any 1000 k m 2 'area' in this ecoregion, the analyses of Malamud et al. (2005) would indicate on average one wildfire with burned area greater than 10 kill 2 every 1 - 3 years,

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Fig. 3. Spatial mapping of wildfire recurrence intervals for the conterminous USA by ecoregion division (figure after Malamud et al. 2005). Based on a power-law frequency-area relationship from empirical data for the period 19702000 (see Fig. 2), recurrence intervals show how many years on average a wildfire of 10 km2 or larger would be expected in spatial areas of 1000 km2 within each ecoregion division. The legend colours go from black (small recurrence intervals) to white (large recurrence intervals), representing 'high' to 'low' hazard, with the legend scale in years increasing logarithmically.

or 3 3 - 1 0 0 % probability of occurring in any given year. This is compared to the W a r m C o n t i n e n t a l ecoregion, where there is a 0.3-1.0% probability for the same size wildfire (10 km 2 or greater) occurring in any given year. Even with the large error bars, these results are useful for broad generalities of wildfire risk in the conterminous USA. The question is whether they are useful for accurate probabilistic hazard forecasting on a finer geographic scale compared with more complex models that might provide more accurate assessment of the assessment at the cost of needing greater parameterization (e.g. Presiler et al. 2004). However, not all scientific models must be used for prediction, and Malamud et al. (2005) argue that the two-parameter 'parsimonious' power-law model is a useful tool to learn about wildfire regimes at broad, regional scales as other authors have recently emphasized (Cleland et al. 2004; Schoennagel et al. 2004). These studies do not lend themselves easily to applied wildfire management in terms of specific probabilistic hazard estimation or wildfire prediction (i.e. when and where a fire will occur) at specific points in the landscape. Rather these studies are useful to examine the behaviour of

wildfire regimes at broader scales - for broader management issues over longer time-scales and larger spatial extents. However, the examination of the driving forces and most important processes between regions increases understanding of these systems and will become increasingly useful for management. Describing wildfire regimes through the simplified assumption of power-law behaviour will be one aspect of this examination of the driving forces behind wildfire regimes. Finally, there are possible links between other more complicated statistical tools and the simple power-law approach outlined here. Katz et al. (2005, p. 1133) comment that 'an apparently unappreciated connection between the existence of power laws in ecology and statistical extreme event theory has been identified'. Although the usual application of statistics emphasizes the mean and variance as a probability distribution's parameters of interest, extreme event statistics (and extreme value theory) focus on a variable's extremal values (Gaines & Denny 1993; Katz et al. 2005). Moritz (1997) compares the 'extremal fire regime' (i.e. the distribution of the largest wildfires in each year) in two regions of the Los Padres National Forest (California, USA) in relation to

QUANTIFYING WILDFIRE REGIMES wildfire suppression and climatic forcing events. Although seemingly comparatively infrequently used in fire regime studies, the work of Moritz (1997) suggests that the approach has potential utility for comparison of wildfire regimes over broad space-time scales, and also for exploring the relative importance of different forcing mechanisms (by including other environmental factors as covariates in models of the extremal wildfire regime). A similar approach has been taken by Alvarado et al. (1998). Gaines & Denny (1993) also comment on the observed spatial consistency of parameter estimates of extreme value distribution, and go so far as to consider that this may be indicative of the 'existence of underlying principles governing these phenomena' (p. 1677). The question of what drives and/or constrains spatiotemporal variability in model coefficients and observed probability distributions remains to be adequately addressed.

Conclusion Quantitative description of disturbance regimes is of considerable interest to ecologists and others, from both theoretical and applied standpoints. Wildfire frequency-area distributions have received ongoing attention. On the one hand ecologists are interested in unravelling the importance of 'extreme' events for ecosystem composition and function, on the other there has been considerable interest in the idea that power-law frequency-area wildfire distributions indicate the presence of 'self-organized criticality'. From an applied perspective there is an obvious interest in being able to predict the likelihood of extreme events for hazard management and mitigation. Although heavy-tailed distributions typify observed wildfire frequency-area distributions, there is considerable debate over the exact nature of the probability distribution(s) that best describe these data; some authors have strongly advocated power laws as best descriptors while others have not. This debate has been muddied by interpretations of systems being self-organized critical depending on what type of distribution is observed. Although power laws provide a simple and parsimonious description of many observed frequency-area distributions, more caution needs to be taken in ascribing the presence of power laws to a system being in the self-organized critical state than has often been the case. Studies examining the 'Why?' and 'What use?' of power laws in nature should be extended in the future. Irrespective of theoretical debates regarding the complexity theory underlying wildfire distributions, or regarding the distributions themselves, future studies should utilize techniques associated with self-organized criticality, cellular

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automata modelling and statistical physics to build bridges toward the ecological community. In turn, this will allow them to become of greater value as tools for examining ecological processes and systems, while attempting to improve understanding of fundamental underlying natural laws. The contributions of author B.D.M. were partially supported by the UK NERC/EPSRC Grant NER/T/S/ 2003/00128.

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GUICHARD, F., HALPIN, P. M., ALLISON, G. W., LUBCHENCO, J. & IVIENoE,B. A. 2003. Mussel disturbance dynamics: signatures of oceanographic forcing from local interactions. American Naturalist, 161, 889-904. HEINSELMAN, M. L. 1973. Fire in the virgin forests of the Boundary Waters Canoe Area, Minnesota. Quaternary Research, 3, 329-382. HENLEY, C. L. 1993. Statics of a 'self-organized' percolation model. Physical Review Letters, 71, 2741-2744. HERGARTEN, S. 2002. Self-Organized Criticality in Earth Systems. Springer, Berlin. HOLMES, T. P., PRESTEMON, J. P., PYE, J. M., BUTRY, D. T., MERCER, D. E. & ABT, K. L. 2004. Using size-frequency distributions to analyze fire regimes in Florida. In: ENGSTROM, R. T., GALLEY, K. E. M. & DE GROOT, W. J. (eds) Pro-

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QUANTIFYING WILDFIRE REGIMES MINNICH, R. A. 1983. Fire mosaics in Southern California and Northern Baja California. Science, 219, 1287-1294. MINNICH, R. A. 2001. An integrated model of two fire regimes. Conservation Biology, 15, 1549-1553. MIYANISHI, K. & JOHNSON, E. A. 2001. Comment: a re-examination of the effects of fire suppression in the boreal forest. Canadian Journal of Forest Research, 31, 1462-1467. MORITZ, M. A. 1997. Analyzing extreme disturbance events: fire in Los Padres National Forest. Ecological Applications, 7, 1252-1262. NIKLASSON, M. & GRANSTROM, A. 2000. Numbers and sizes of fires: long-term spatially explicit fire history in a Swedish boreal landscape. Ecology, 81, 1484-1499. ORESKES, N., SHRADER-FRECHETTE,K. & BELITZ, K. 1994. Verification, validation, and confirmation of numeric models in the earth sciences. Science, 263, 641-646. PASCUAL, M. & GUICHARD, F. 2005. Criticality and disturbance in spatial ecological systems. TRENDS in Ecology and Evolution, 20, 88-95. PERRY, G. L. W. 1998. Current approaches to modelling the spread of wildland fire: a review. Progress in Physical Geography, 22, 222-245. PERRY, G. L. W. 2002. Landscapes, space and equilibrium: shifting viewpoints. Progress in Physical Geography, 26, 339-359. PRESILER, H. K., BRILLINGER,D. R., BURGAN, R. E. & BENOIT, J. W. 2004. Probability-based models for estimation of wildfire risk. International Journal of Wildland Fire, 13, 133-142. REED, W. J. & HUGHES, B. D. 2002. From gene families and genera to incomes and internet file sizes: why power laws are so common in nature. Physical Review E, 66, Article Number 067103. REED, W. J. & MCKELVEY, K. S. 2002. Power-law behaviour and parametric models for the sizedistribution of forest fires. Ecological Modelling, 150, 239-254. REED, W. J. & JOHNSON, E. A. 2004. Statistical methods for estimating historical fire frequency from multiple fire-scar data. Canadian Journal of Forest Research, 34, 2306-2313. RICOTTA, C., AVENA, G. & MARCHETTI,M. 1999. The flaming sandpile: self-organized criticality and wildfires. Ecological Modelling, 119, 73-77. RICOTTA, C., ARIANOUTSOU, M. ET AL. 2001. Selforganized criticality of wildfires ecologically revisited. Ecological Modelling, 141, 307-311. ROMME, W. H., EVERHAM, E. n., FRELICH, L. E., MORITZ, M. A. & SPARKS, R. E. 1998. Are large, infrequent disturbances qualitatively different

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from small, frequent disturbances? Ecosystems, 1, 524-534. SCHENK, K., DROSSEL, B., CLAR, S. & SCHWABL, F. 2000. Finite-size effects in the self-organized critical forest-fire model. European Physical Journal B, 15, 177-185. SCHOENBERG, F. P., PENG, R. & WOODS, J. 2003. On the distribution of wildfire sizes. Environmetrics, 14, 583-592. SCHOENNAGEL, T., VEBLEN, T. Z. & ROMME, W. H. 2004. The interaction of fire, fuels, and climate across rocky mountain forests. Bioscience, 54, 661-676. SOLOW, A. R. 2005. Power laws without complexity. Ecology Letters, 8, 361-363. SONG, W. G., FAN, W. C., WANG, B. H. & ZHOU, J. J. 2001. Self-organized criticality of forest fire in China. Ecological Modelling, 145, 61-68. SPURR, S. H. 1954. The forests of Itasca in the nineteenth century as related to fire. Ecology 35, 21-25. STRAUSS, D., BEDNAR, L. & MEES, R. 1989. Do one percent of forest fires cause ninety-nine percent of the damage? Forest Science, 35, 319-328. TURCOTTE, D. L. 1999. Self-organized criticality. Reports on Progress in Physics, 62, 1377-1429. TURCOTTE, D. L. & MALAMUD, B. D. 2004. Landslides, forest fires, and earthquakes: examples of self-organized critical behaviour. Physica A, 340, 580-589. TURNER, M. G., BAKER, W. L., PETERSON, C. J. & PEET, R. K. 1998. Factors influencing succession: lessons from large, infrequent natural disturbances. Ecosystems, 1, 511-523. WARD, P. C., TITHECOTT, A. G. & WOTTON, B. M. 2001. Reply: A re-examination of the effects of fire suppression in the boreal forest. Canadian Journal of Forest Research, 31, 1467-1480. WHITE, P. S. & PICKETT, S. T. A. 1985. Natural disturbance and patch dynamics: An introduction. In: PICKETT, S. T. A. & WHITE, P. S. (eds) The

Ecology of Natural Disturbance and Patch Dynamics. Academic Press, New York, 3-13. Wu, J. & LOUCKS, O. L. 1995. From balance of nature to hierarchical patch dynamics: a paradigm shift in ecology. Quarterly Review of Biology, 70, 437 -466. YANG, C. B. 2004. The origin of power-law distributions in self-organized criticality. Journal of Physics A, 37, L523-L529. ZHANG, Y. H., WOOSTER, M. J., TUTUBALINA, O. & PERRY, G. L. W. 2003. Monthly burned area and forest fire carbon emission estimates for the Russian Federation from SPOT VGT. Remote Sensing of Environment, 87, 1 - 15.

Index Acoustic emission (AE), 11-29, 47-61, 63-78 advantages and disadvantages, 11 case histories, 56-59 crustal stress, fractal analysis, and 47-61, 63-78 energy release rate, granitic rocks, 11-29 Greece, 63-78 Italy, 47-61 techniques, 71-76 Active faults, b-value and fractal dimension, 26, 133-140 AE. See acoustic emission Agia Efimia area, Kefallinia, Greece, 70 Ainos block, Greece, 64-66 Apennines, Italy, 50-51, 55, 113, 115, 133-136, 139 grabens features of Miocene Early Pleistocene age, 50 historical earthquakes, 137 map of active faults, 136 Argostoli block, Greece, 64 Asteroid impacts, 1 b-value. See also earthquakes, Gutenberg-Richter, seismicity estimating, 15 fractal dimension, 1-2, 25-26, 133-140 granitic rocks pre-failure damage experiments and, 14-15 Italy, 134 recurrence time, 15-16 southern California, USA, 2 stress in rocks, 11-29 variations and significance, 24 Bed thickness, fracture spacing and, 126-127 Biodiversity, and intermediate disturbance hypothesis, 156 CA. See cellular-automata Cascade of clusters, 1-9 Cellular automata (CA) models, 4 - 5 avalanche behaviour, 4 - 5 forest-fire model, 1, 5-8, 155, 157-159, 161-162 frequency-size statistics, 1, 4 - 8 inverse-cascade model, 5 - 8 metastable regions, 1, 5-6, 8 power-law frequency-area statistics, 1, 4 - 8 sandpile model, 1, 4-5, 157 slider-block model, 5 Charles's power law, 19-20 Cheat River, West Virginia, USA, flood analysis, 143, 147 Chow's theory of multiplicative processes, 141 Clusters cascade of clusters, small to large, 1-9 crustal stress and seismicity, Greece, 67, 71 faults, fractures, joints, 33-34, 37, 96, 116-118, 121,124, 129 growth, in forest-fire model, 1-9, 161 percolation, 157 Colfiorito seismogenic zone, Italy, 48, 56, 135, 137, Complexity theory in nature, and power-law relation, 96, 139, 141-142, 165

Connectivity, multiscale fracture networks and, 31-45, 113, 115, 121, 126-128 Correlation length, 34, 39, 161 Correlogram, 81, 84, 90-91 Cracks in rocks, 11-29, 31-45, 53, 80, 85, 119. See also microcracking Critical. See also self-organized criticality critical path and bond, in flow, 39 critical phenomena, 4, 7, 12, 80, 86, 90-91, 133, 155, 157, 163, 165 path analysis, 39 subcritical crack growth, 11-12, 14, 18-23, 27 Crustal stress AE techniques, 71-76 crises, 47-61, 63-77 DEM techniques, 70-71 DInSAR techniques, 66-70 GPS techniques, 64-66 Ionian archipelago, Greece, 63-77 Italian peninsula, 47- 61 soil exhalation, 71-76 world stress map, 53-54 Damage localization Charles's power law, 19 failure nucleation, 24-25 pre-failure damage, granitic rocks, 11-29 DATAPLORE Software, rock slope displacement, 85 DEM. See digital elevation model Detrended fluctuation analysis (DFA), 85-86, 89, 96-97, 99, 100 DFFN. See discrete fault and fracture model DGPS. See Differential Global Position System Differential Global Position System (DGPS), 63-64, 66, 71 Differential Interferometric Synthetic Aperture Radar (DInSAR), 64, 66 Digaleto area, Kefallinia, Greece, 70 Digital Elevation Model (DEM), techniques, 70-71 Dilatometric measurements of cracks, 80, 85-90 DInSAR. See Differential lnterferometric Synthetic Aperture Radar Discrete fault and fracture model (DFFN), 113, 115, 128-129 Disturbance regimes, wildfires, 155-156, 159, 165 Dual porosity model, fractures, 128 Dynamical systems theory, 79-81, 157-158 Earthquakes. See also acoustic emission, b-value, crustal stress crises, faults, Gutenberg-Richter, seismic, slider-block model body-wave magnitude, 14 fault zones and fractal dimension, 25, 133-139 foreshocks, 24 frequency-size statistics, 1-2 geoelectrical time series analysis of, 95-103 Greece, 63-78 Italy, 47-61, 95-103, 133-140 landslides triggered by, 3, 105-106 Omori law, 24

From: CELLO, G. & MALAMUD,B. D. (eds) 2006. Fractal Analysis for Natural Hazards.

Geological Society, London, Special Publications, 261, 169-172. 0305-8719/06/$15.00 9 The Geological Society of London 2006.

170 Earthquakes (Continued) prediction of via b-value, 26 USA, 2, 106 Ecology and ecosystems, succession-disturbance theory, and wildfires 156-157 Elasticity, concept of, 51-52 Emergent simple scaling, flooding analysis, 142-144, 151 Energy release rate and AE magnitude, 14-17, 21-23, 26-27 Erissos peninsula, Greece, 64 Failure nucleation, damage localization and, 24-25 Faults. See also acoustic emissions, damage localization, earthquakes, fractures bioclastic limestone, 116-117 Apennines fault system, Italy, 133-140 Caramanico fault system, Italy 115-116 earthquake magnitude and fractal dimension, 133-139 fault seal analysis, 113 Kefallin]a transform fault, Greece, 66, 72 Majella Mountains, Italy, 113-131 microcracking phases, 11 Nojima fault zone, Japan, 13, 25 Anatolian fault zone, Italy, 51, 65 porous grainstones, 116 San Andreas fault system, USA, 33 spacing, 117-118 spatial pattern as predictor of earthquake size, 133-139 FFA. See flood frequency analysis Firing frequency, forest fire model, 5, 7, 158-159 Flood frequency analysis (FFA), 1, 141-153 Log-Pearson III (LP3), 142, 147-149 magnitude/frequency curves, 143-144 Mississippi River study, 143 peak annual discharge data, 144 power-law (PL) model, 142-143 rainfall data, 143, 148-149, 151 strengths and weaknesses, 141-142 US catchment study, PL vs. LP3 models, 148 Flow models, for multiscale networks, 31-45 Fluid transfer, in fractured rock, modelling of, 31 Forecasting, probabilistic hazard damage model, granite, and fractures, 11-29 earthquakes and acoustic emission, 47-61, 63 -78, earthquakes and spatial fault patterns, 133-140 floods, 141-153 rock slopes and rock falls, 79-93, 107-108 wildfires, 163-165 Forest fires. See wildfires Forest-fire model, 1, 5-8, 155, 157-159, 161-162 Fractal. See also heavy-tail, Mandelbrot, multifractal, nonlinear, power-law acoustic emission data, 13, 16-19, 24-27, 47-61, 63-78 band-limited, 24 bed thicknesses, 127 box counting method, 59-60 b-value, relationship to fractal dimension, 1-2, 25-26, 133-140 clusters, spatially, 1, 6-8, 36-37

INDEX construction, 7-8 crack populations, 11-12 fault zones, 25, 133-139 floods, 1, 141-153 fracture networks, 31-45, 113-131 geoelectrical data, multifractal variability, 95-103 landslide events, 105-111 natural hazards, in general, 1-4 rock slopes and rock falls, 79-93, 107-108 seismogenic faults, 133-140 self-affine, 31, 91 self-similar, 1, 33, 59, 141-143, 151,157, 163 soil exhalation, 47-61, 71-76 wildfires, 8, 155-167 Fractures. See also faults aperture distributions, 31 bed thickness and, 127 clustering, fractal networks and, 36-37 connectivity and, 35-36 damage phase, 22-24 DFFN model, 128, 129 dual porosity model, 128 dynamics, study techniques, 11 Hornelen basin, 33 hydrothermal fluid migration, 31 length distributions, 31, 39 length, three-dimensional rules, 35 Majella Mountains, Italy, 113-131 mechanism, 22-24 network connectivity, 33-34 percolation theory, permeability, 33-34 permeability as constant, 39-40 scale invariance of, 32 transmissivity, 37-38 Fragmentation, rockfalls and, 108 Frequency-area statistics. See frequency-size statistics Frequency-size statistics. See also Charles's law, Gutenberg-Richter, heavy-tail, inverse-gamma, power-law, self-similar asteroid impacts, 1 bed thickness, 127 cellular-automata models, 4-5, 157-159 crack populations, 11 - 12 earthquakes, 1-2, 26 floods, 1, 141-153 fractures, 32-34, 117-118, 120-122 inverse cascade, and, 1-9 landslides, 3, 105-111 rockfalls, 107-108 volcanic eruptions, 1 wildfires, 3-4, 155-167 Geoelectrical data, Giuliano, Italy, 95-103 Giuliano, Italy data, 48, 55-56, 96-99 Granite, experiments on, 11-27 Graphical tools, phase space portraits and correlograms, 81- 86 Greece, crustal stress and seismicity, 63-77 Ground deformation. See Differential Interferometric Synthetic Aperture Radar Gutenberg-Richter frequency-magnitude relation, 1, 2, 15-16, 26, 105, 108

INDEX Heavy-tail frequency-size statistics. See also frequency-size statistics, power-law rainfall, 148 wildfires, 155, 157, 159, 161,162, 165 Hellenic arc, trench system, 64-65 Hidden information, nonlinear science issues and rock slopes, 80-86 Highly Optimized Tolerance (HOT), 162 Histograms. See frequency-size distributions Htlder exponent, 95, 100-102 HOT. See Highly Optimized Tolerance Hurst exponent, 95, 98-100, 102 Hydraulic models, general, 38

Natural hazards. See asteroid impacts, earthquakes, floods, landslides, rockfalls, volcanoes, wildfires Non-fractal networks, 34-36 Nonlinearity. See also fractal, multifractal, self-organized criticality acoustic emissions, granitic rocks, and, 14, 21, 23 definition of, 80 sensu stricto, definition of, 80 unstable rock slopes, 79-93 Nonstationary, 95, 97 Nucleation phase, 11, 16-27

Intermediate disturbance hypothesis, 156 Inverse-cascade model, 1-9 Inverse-gamma distribution, landslides, 3, 105-109 Italy crustal stress crises and seismic activity in, 47-61 fracture systems, Majella mountain region, 113-131 landslides, Todi and Umbria, 3, 105-111 multifractal variability in self-potential signals, 95-103 seismogenic faults, central Apennines, 133-140 tectonic framework, 47-51 wildfires, 160

Orchi, Italy, acoustic emissions, 57

Kefallin]a, Greece, 58, 63-77 Lacunarity, definition of, 37 Landslides. See also rockfalls area, average in triggered events, 108 frequency-size distributions, 1, 3, 105-108 'general' probability distribution, 3, 105-111 Guatemala, 3, 106, 108 historical/incomplete landslide inventories, 109 inverse-gamma distribution, 3, 106-107 Italy, 3, 105-111 magnitude scale, 108-109 power-law frequency-area statistics, 3, 107 triggers of, 3, 105 USA, 3, 105-106, 108 Lattice-based models. See cellular-automata (CA) models Lefkada earthquake, 64, 66-67, 76 Limestone, 116-117, 122 Log-Pearson III (LP3), t41-143, 145-146, 151 LP3. See Log-Pearson III Magnitude-frequency curves. See frequency-size statistics Majella Mountains, Italy, fracture study, 113-131 Mandelbrot, vii (preface), 32, 137, 142 Mediterranean ecoregion, USA, wildfire statistics, 4, 8, 159, 161, 163 Metastable region, 5-6, 8 Microcracking, 11, 13-14, 16-17, 22-23, 25, 27 Molise earthquake, Italy, 48, 55-58 Multifractals Detrended Fluctuation Analysis (DFA), 97 floods, power-laws, and, 142-143 pre-failure damage, and, 16, 25 self-potential signals, 95-103 Multiscale fractal fracture networks, 31-45

Peaks-over-threshold, floods, 143, 150 Percolation, 33-39, 43, 157 Permeability, flow in fracture networks, 31-45 Phase portrait, rock slope displacement analysis, 81, 83, 91 Phase transition, floods, 143-144, 150 PL. See power-law. Power-law frequency-size distribution. See also fractal, frequency-size distributions, Gutenberg-Richter, heavy-tail, self-similar asteroid impacts, 1 bed thicknesses, 127 cellular-automata models, 1, 4-5, 157-159 Charles's power law, 19-20 crack populations, 11 - 12 earthquakes, 1-2, 26 floods, 1, 141-153 fractures, 32-34, 117-118, 120-122 Highly Optimized Tolerance (HOT), 162 inverse cascade, and, 1-9 landslides, 3, 105-111 mechanisms of, 163 rockfalls, 107-108 volcanic eruptions, 1 vs. log-Pearson III distribution, flood frequency analysis, 147-149 wildfires, 3-4, 155-167 Prediction. See forecasting Probabilistic hazard assessment. See forecasting Rainfall floods, relation to, 141-143, 148, 151 landslides, trigger for, 3, 105-109 Raponi, Italy, acoustic emissions, 56-57 Recurrence time. See also forecasting b-value, relation to, 15 granitic rocks pre-failure damage experiments, 15-16 wildfires, 162-164 Rockfalls, 107-108. See also rocks Rocks cracks, 11-29, 31-45, 53, 80, 85, 119 failure, predictability of, 26-27 fracturing, 14, 16-19, 22-23, 25-26 near to stability plots, of rock slope displacement, 82 rock slope displacement analysis, 79-91

171

172 Sandpile model, 1, 4-5 Scaling. See fractal, self-similar, power-law Seismic activity. See earthquakes Self-affine distribution, 31, 91 Self-organized criticality (SOC) 4, 7, 113, 133, 155, 157-159, 162-163, 165 Self-potential signals, seismic areas, 95-103 Self-similar cascade of clusters from small to large, 1-9 distribution, identification, 59, 143-144 Singularity spectra, Giuliano, Italy series, 100-102 Slider-block model, 1, 5 Slope failures, 79-93 SOC. See self-organized criticality Software Bernes GPS software, 65 Dataplore software, 85 static kinematic software, 64 WINFAP, flood software, 145 Soil exhalation Italy study, fractal analysis of, 47-60 Kefallinia, Greece study, 63-78 Sparking frequency, forest fire model, 5, 7, 158-159 Spectral analysis, in self-potential dynamics, 96 Static Kinematic Software, 64 Stress crises. See crustal stress, crises Subtropical ecoregion, wildfire power-law statistics, 4, 8, 159, 161 Succession, in ecology, definition of, 156 Tensile stress, mechanisms for, 23 Todi, Italy, landslides, 107-109 Tyrrhenian seafloor, 48-52

INDEX USA earthquakes, 2-3, 105-106, 108 flood catchment study, 141-153 landslides, 3, 105-106, 108 wildfires, 3-4, 8, 155-167 Volcanoes, 1, 53, 58, 65, 108 Wildfires. See also forest-fire model Australia, 4, 160 Canada, 160-161 China, 160 Europe (France, Greece, Italy, Spain, Sweden), 160 disturbance regimes, 155-156, 159, 165 forecasting, power-law analysis for, 165-166 frequency-area distributions, 156-160 heavy-tailed wildfire frequency-area distributions, 159, 161-162 Mediterranean ecoregion, 4, 8, 159, 161, 163 meteorological conditions and, 4 power-law scaling in, 3-4, 159 recurrence intervals map, USA, 164 self-organized criticality (SOC), 162-163 subtropical ecoregion, 4, 8, 159, 161 succession-disturbance theory, 156-157 Russia, 160 USA, 3-4, 8, 155-167 World Stress Map, 53-54

Fractal Analysis for Natural Hazards Edited by G. Cello and B. D. Malamud

In the Earth sciences, the concept of fractals and scale invariance is well recognized in many natural objects. However, the use of fractals for spatial and temporal analyses of natural hazards has been less used (and accepted)in the Earth sciences. This book brings together 12 contributions that emphasize the role of fractal analyses in natural hazard research, including landslides, wildfires, floods, catastrophic rock fractures and earthquakes. A wide variety of spatial and temporal fractal-related approaches and techniques are applied to 'natural' data, experimental • ... ~.~..... .idata and computer simulations. These approaches include probabilistic hazard analysis, cellular-automata models, spatial analyses, temporal variability, prediction and self-organizing behaviour. The main aims of this volume are (a) to present current research on fractal analyses as applied to natural hazards and (b) to stimulate the curiosity of advanced Earth science students and researchers in the use of fractals analyses for the better understanding of natural hazards.

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Phanerozoic Ironstones (Geological Society Special Publication)

Slope Tectonics (Geological Society Special Publication 351)

Palaeozoic Reefs and Bioaccumulations: Climatic and Evolutionary Controls - Special Publication no 275 (Geological Society Special Publication)

Devonian Events and Correlations : Special Publication no 278 (Geological Society Special Publication)

Non-Marine Permian Biostratigraphy and Biochronology - Special Publication No. 265 (Geological Society Special Publication)

Structurally Complex Reservoirs - Special Publication no 292 (Geological Society Special Publication)

Building Stone Decay: From Diagnosis to Conservation - Special Publication no 271 (Geological Society Special Publication)

Confined Turbidite Systems (Geological Society Special Publication No. 222)

Deformation of Glacial Materials (Geological Society Special Publication No. 176)

Volcanic Degassing (Geological Society Special Publication No. 213)

Fractured Reservoirs (Geological Society Special Publication no 270)

Geomaterials in Cultural Heritage (Geological Society Special Publication No. 257)

Hydrothermal Vents and Processes (Geological Society Special Publication No. 87)

Climates: Past and Present (Geological Society Special Publication No. 181)

Hydrocarbons in Crystalline Rocks (Geological Society Special Publication No. 214)

The Evolving Continents: Understanding Processes of Continental Growth - Special Publication 338 (Geological Society Special Publication)

Elevation Models for Geoscience: Geological Society Special Publication 345

History of Palaeobotany: Selected Essays (Geological Society Special Publication)

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History of Palaeobotany: Selected Essays (Geological Society Special Publication)

Palaeoproterozoic Supercontinents and Global Evolution (Geological Society London, Special Publication)

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